#6456340

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1953 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363536);