#7604436

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1861 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)(5,6,9,8)(7,10,12,11)(14,20)(15,23)(16,18)(17,21)(22,24)(25,28);(2,3)(5,7)(6,10)(8,11)(9,12)(13,14,18,20,19,24,16,22)(15,25,26,17,23,21,27,28);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,15)(14,21)(16,27)(17,22)(18,26)(19,23)(20,28)(24,25);(1,3,4,2)(7,12)(10,11)(13,16,19,18)(14,22,24,20)(15,26,23,27)(17,21,28,25);(13,17,16,25,19,28,18,21)(14,23,22,26,24,15,20,27)

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=h,d^2=g,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=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, 7511632);