#7608651

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	#2416	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(20,27)(24,31)(26,29)(28,32);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18,22,25)(19,23,21,30)(20,28,27,32)(24,26,31,29);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,19,22,21)(18,23,25,30)(20,29,27,26)(24,32,31,28);(10,14)(13,16)(17,20,19,26,22,27,21,29)(18,24,30,32,25,31,23,28);(2,5)(6,8)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,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,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, 7515847);