#244537

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

1025 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1025 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) = 64

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

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

Polycyclic presentation:

2^9|b^2=i,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, 151733);