#6382130

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1954 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)(3,4)(6,10)(11,12)(13,14,19,15,18,22,16,20)(17,21,27,23,26,28,24,25);(5,6)(7,11)(8,12)(9,10)(13,15,19,22,18,20,16,14)(17,25,27,21,26,23,24,28);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,19)(14,20,22,15)(17,24,26,27)(21,25,28,23);(5,8,9,7)(6,11,10,12)(13,17,18,26)(14,21,22,28)(15,23,20,25)(16,24,19,27);(17,26)(21,28)(23,25)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6289326);