#6467155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437415
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ 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:

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

Polycyclic presentation:

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

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, 6374351);