#7503932

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1994 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411128);