#6621086

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

1786 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)(13,14,19,23)(15,25,16,22)(17,24,18,21)(20,27,28,26);(1,2,3,4)(6,10)(8,11)(13,15)(14,20)(16,19)(17,27)(18,26)(21,22)(23,28)(24,25);(5,6)(7,8)(9,10)(11,12)(14,21)(15,16)(17,18)(20,25)(22,28)(23,24);(5,7,9,12)(6,11,10,8)(13,16,19,15)(14,22,23,25)(17,27,18,26)(20,24,28,21);(5,8,9,11)(6,7,10,12)(14,23)(15,16)(21,24)(26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,d^2=h,e^2=hi,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, 6528282);