#245341

order	512
nilpotency class	3
isoclinism	isoclinic class of #245303
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	C₂×C₄²	C₂⁴
prime radical	C₂³	#534
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

890 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152537);