#7608156

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	#28921	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1387 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=ghi,d^2=gi,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di

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, 7515352);