#6621076

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1795 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)(14,19)(15,22)(17,27)(21,28)(23,24)(25,26);(1,2,3,4)(6,10)(8,11)(13,14,18,21)(15,16,24,20)(17,28,27,19)(22,25,23,26);(5,6)(7,8)(9,10)(11,12)(13,15,18,24)(14,20,21,16)(17,23,27,22)(19,26,28,25);(5,7,9,12)(6,11,10,8)(13,16,18,20)(14,15,21,24)(17,26,27,25)(19,22,28,23);(5,8,9,11)(6,7,10,12)(13,17,18,27)(14,19,21,28)(15,23,24,22)(16,25,20,26)

Polycyclic presentation:

2^9|b^2=fg,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528272);