#6467672

order	512
nilpotency class	3
isoclinism	isoclinic class of #24269
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,21)(15,20,16,19)(18,22,24,27)(23,28,25,26);(1,2,3,4)(15,23)(16,25)(18,24)(19,26)(20,28)(22,27);(6,9)(10,12)(13,15,18,23,17,16,24,25)(14,19,22,26,21,20,27,28);(13,16,18,25,17,15,24,23)(14,20,22,28,21,19,27,26);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374868);