#6454528

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1974 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)(13,14,17,22,18,20,30,34)(15,23,25,31,26,36,29,19)(16,21,28,35,27,33,40,43)(24,37,39,42,38,44,41,32);(1,2)(3,4)(13,15,18,26)(14,19,20,31)(16,24,27,38)(17,29,30,25)(21,32,33,42)(22,36,34,23)(28,41,40,39)(35,44,43,37);(2,4)(6,9)(10,12)(14,20)(16,27)(19,31)(22,34)(23,36)(24,38)(28,40)(39,41);(2,4)(13,16,18,27)(14,21,20,33)(15,24,26,38)(17,28,30,40)(19,32,31,42)(22,35,34,43)(23,37,36,44)(25,39,29,41);(5,7,8,11)(6,10,9,12)(16,27)(21,33)(24,38)(28,40)(32,42)(35,43)(37,44)(39,41)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6361724);