#6529744

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1745 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436940);