#7605792

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgi,c^2=h,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7512988);