#6487705

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470137
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1970 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6394901);