#7614736

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#24370	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1613 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,22)(23,28)(24,29)(30,31);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(18,22)(19,25)(21,26)(23,30)(27,32)(28,31);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,18)(19,23)(20,22)(21,24)(25,31)(26,29)(27,28)(30,32);(10,14)(13,15)(17,19,26,32,20,27,21,25)(18,23,29,30,22,28,24,31);(2,5)(6,8)(10,14)(13,15)(18,22)(19,27)(24,29)(25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521932);