#6814463

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1839 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)(7,11)(8,10)(9,12)(13,14)(15,25)(16,29)(17,32)(18,22)(19,24)(20,27)(21,28)(23,33)(26,35)(30,36)(31,37)(34,44)(38,43)(39,40)(41,42);(5,7)(6,9)(8,12)(10,11)(13,15,18,27)(14,20,22,25)(16,30,28,19)(17,31,33,41)(21,36,29,24)(23,37,32,42)(26,34,40,43)(35,38,39,44);(1,2,3,4)(13,16,19,15,18,28,30,27)(14,21,24,20,22,29,36,25)(17,31,34,26,33,41,43,40)(23,37,38,35,32,42,44,39);(6,10)(9,11)(14,22)(17,33)(20,25)(21,29)(24,36)(26,40)(31,41)(34,43);(7,12)(9,11)(13,17,19,34,18,33,30,43)(14,23,24,38,22,32,36,44)(15,26,28,41,27,40,16,31)(20,35,29,42,25,39,21,37)

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6721659);