#7616324

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	C₂×#127	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#127 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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)(4,5)(6,8)(11,12)(14,15)(17,18)(19,28)(20,30)(21,32)(22,25)(23,27)(24,29)(26,31);(2,5)(6,7)(9,10,13,16)(11,15,12,14)(17,19,22,29)(18,24,25,28)(20,23,31,21)(26,27,30,32);(9,11,13,12)(10,14,16,15)(18,25)(19,29)(20,31)(27,32);(9,12,13,11)(10,15,16,14)(17,20,23,29,22,31,21,19)(18,26,27,28,25,30,32,24);(1,3,4,8)(2,6,5,7)(17,21,22,23)(18,27,25,32)(19,20,29,31)(24,30,28,26)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=h,e^2=f,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^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, 7523520);