#6375794

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

1889 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6282990);