#6375774

order	512
nilpotency class	3
isoclinism	isoclinic class of #60756
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 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)(6,9)(7,11)(14,16)(15,21)(18,25)(19,27)(22,28)(23,26);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,22,23,19)(18,20,25,24)(21,27,26,28);(6,9)(10,12)(13,15,17,23)(14,19,16,22)(18,26,25,21)(20,27,24,28);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,22,23,19)(18,24,25,20)(21,27,26,28);(14,16)(19,22)(20,24)(27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6282970);