#7600412

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	#25123	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 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)(4,6)(5,8)(17,18,24,27)(19,28,29,32)(20,30,31,25)(21,26,23,22);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,30,24,29,23,25)(18,20,22,32,27,31,26,28);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(17,20,24,31)(18,25,27,30)(19,26,29,22)(21,32,23,28);(2,6)(5,7)(10,14)(13,16)(17,21,24,23)(18,22,27,26)(19,25,29,30)(20,28,31,32);(2,6)(5,7)(17,22,23,18,24,26,21,27)(19,28,25,31,29,32,30,20)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=i,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, 7507608);