#7611932

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#45332	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2031 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=i,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei

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, 7519128);