#7501992

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

1847 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,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, 7409188);