#6467793

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1828 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)(13,14,18,22)(15,23,25,38)(16,20,29,36)(17,31,24,21)(19,32,33,27)(26,39,28,42)(30,44,40,37)(34,43,35,41);(2,3)(13,15,24,32,18,25,17,27)(14,19,31,38,22,33,21,23)(16,28,40,41,29,26,30,43)(20,35,44,39,36,34,37,42);(1,2,4,3)(6,9)(10,12)(15,25)(19,33)(23,38)(26,28)(27,32)(34,35)(39,42)(41,43);(1,3,4,2)(13,16,18,29)(14,20,22,36)(15,26,25,28)(17,30,24,40)(19,34,33,35)(21,37,31,44)(23,39,38,42)(27,41,32,43);(5,7,8,11)(6,10,9,12)(16,29)(20,36)(26,28)(30,40)(34,35)(37,44)(39,42)(41,43)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374989);