#6621080

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1755 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)(13,14,18,22)(15,23,26,27)(16,21,17,19)(20,24,28,25);(1,2,3,4)(6,10)(8,11)(14,19)(16,17)(20,27)(21,22)(23,28)(24,25);(5,6)(7,8)(9,10)(11,12)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,23)(22,28);(5,7,9,12)(6,11,10,8)(13,16,18,17)(14,21,22,19)(15,24,26,25)(20,23,28,27);(5,8,9,11)(6,7,10,12)(14,22)(19,21)(20,28)(23,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=hi,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, 6528276);