#7535517

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1542 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,22,21)(19,23,27,26)(20,24,28,29)(25,30,32,31);(2,6)(7,8)(11,16)(13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,32)(29,31);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,25)(21,29)(22,28)(23,30)(26,31)(27,32);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,21)(19,26)(23,27)(24,29)(25,31)(30,32);(1,4,5,3)(2,7,6,8)(19,27)(20,28)(23,26)(24,29)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281945
maximal normal subgroup of	#72281945

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7442713);