#6487735

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1920 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)(13,14,19,23)(15,24,27,34)(16,21,29,38)(17,22,18,31)(20,32,35,26)(25,39,42,44)(28,36,30,37)(33,40,43,41);(1,2,3,4)(7,8)(10,11);(5,7,9,8)(6,10,12,11)(13,15,19,27)(14,20,23,35)(16,25,29,42)(17,26,18,32)(21,33,38,43)(22,34,31,24)(28,40,30,41)(36,39,37,44);(5,8,9,7)(6,11,12,10)(13,16)(14,21)(15,25)(17,30)(18,28)(19,29)(20,33)(22,37)(23,38)(24,39)(26,41)(27,42)(31,36)(32,40)(34,44)(35,43);(6,12)(10,11)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,28,29,30)(20,34,35,24)(21,36,38,37)(25,40,42,41)(33,39,43,44)

Polycyclic presentation:

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

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, 6394931);