#6467791

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	#35276	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1918 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 6374987);