#6382915

order	512
nilpotency class	3
isoclinism	isoclinic class of #61071
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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2071 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)(6,10)(11,12)(13,14,18,16)(15,22,25,28)(17,20,27,21)(19,24,23,26);(1,3,4,2)(5,6)(7,11)(8,12)(9,10)(14,16)(19,23)(20,21)(22,28);(5,7,9,8)(6,11,10,12)(13,15,17,26,18,25,27,24)(14,19,21,22,16,23,20,28);(5,8,9,7)(6,11,10,12)(13,16,18,14)(15,23,25,19)(17,20,27,21)(22,26,28,24);(13,17,18,27)(14,20,16,21)(15,24,25,26)(19,22,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290111);