#6456461

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	#34976	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2114 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)(13,14,17,21,19,22,18,16)(15,23,28,20,27,24,26,25);(1,2)(3,4)(13,15)(14,20)(16,24)(17,26)(18,28)(19,27)(21,23)(22,25);(2,4)(6,9)(10,12)(13,14,17,21,19,22,18,16)(15,24,28,25,27,23,26,20);(2,4)(13,16,18,22,19,21,17,14)(15,25,26,24,27,20,28,23);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,21,22,16)(15,26,27,28)(20,23,25,24)

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=f,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,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, 6363657);