#7610723

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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
prime radical	#23960	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 7517919);