#6621081

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1778 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)(15,23)(16,21)(17,19)(18,22)(20,24)(25,27)(26,28);(1,2,3,4)(6,10)(8,11)(14,19)(16,17)(20,23)(21,22)(24,26)(27,28);(5,6)(7,8)(9,10)(11,12)(13,15)(14,20)(16,24)(17,26)(18,25)(19,23)(21,27)(22,28);(5,7,9,12)(6,11,10,8)(13,16,18,17)(14,21,22,19)(15,24,25,26)(20,27,28,23);(5,8,9,11)(6,7,10,12)(14,22)(19,21)(20,28)(23,27)

Polycyclic presentation:

2^9|b^2=fgi,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, 6528277);