#6467664

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
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

2030 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)(13,14,19,18)(15,20,24,27)(16,17,21,25)(22,23,26,28);(1,2,3,4)(13,15)(14,20)(16,23)(17,26)(18,27)(19,24)(21,28)(22,25);(6,9)(10,12)(13,16,18,25,19,21,14,17)(15,22,20,23,24,26,27,28);(13,17,14,21,19,25,18,16)(15,23,27,22,24,28,20,26);(5,7,8,11)(6,10,9,12)(13,14,19,18)(15,20,24,27)(16,17,21,25)(22,23,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374860);