#244546

order	512
nilpotency class	3
isoclinism	isoclinic class of #244523
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

975 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 975 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,12)(3,13,18,27)(4,19,5,23)(6,24,7,11)(9,17,28,16)(10,21,29,22)(14,31,32,25)(15,26,20,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,49)(44,50)(45,51)(46,59)(47,54)(48,55)(52,57)(53,60)(56,63)(58,62)(61,64);(1,3)(2,9)(4,20)(5,15)(6,17)(7,16)(8,18)(10,26)(11,27)(12,28)(13,24)(14,21)(19,25)(22,32)(23,31)(29,30)(33,35,38,45)(34,39,42,51)(36,43,47,57)(37,44,48,58)(40,52,54,49)(41,50,55,62)(46,56,60,64)(53,63,59,61);(1,4,8,5)(2,10,12,29)(3,14,18,32)(6,21,7,22)(9,25,28,31)(11,23,24,19)(13,30,27,26)(15,16,20,17)(33,36)(34,40)(35,43)(37,46)(38,47)(39,49)(41,53)(42,54)(44,56)(45,57)(48,60)(50,61)(51,52)(55,59)(58,64)(62,63);(1,5,8,4)(2,10,12,29)(3,15,18,20)(6,21,7,22)(9,26,28,30)(11,19,24,23)(13,31,27,25)(14,16,32,17)(33,37)(34,41)(35,44)(36,46)(38,48)(39,50)(40,53)(42,55)(43,56)(45,58)(47,60)(49,61)(51,62)(52,63)(54,59)(57,64)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151742);