#7608139

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1255 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,4)(3,6,8,7)(9,10,14,15)(11,12,16,13)(17,18)(19,27)(20,33)(21,38)(22,39)(23,26)(24,31)(25,35)(28,29)(30,46)(32,44)(34,42)(36,40)(37,43)(41,47)(45,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,29,21,23,32,22,31)(18,24,39,44,26,38,28,27)(20,34,45,47,35,30,37,36)(25,41,48,42,33,40,43,46);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,20)(18,25)(19,30)(21,36)(22,37)(23,35)(24,40)(26,33)(27,42)(28,43)(29,45)(31,47)(32,34)(38,41)(39,48)(44,46);(3,8)(6,7)(9,12,14,13)(10,11,15,16)(18,26)(20,35)(24,38)(27,44)(28,39)(30,34)(36,47)(37,45);(9,13,14,12)(10,11,15,16)(17,21,29,32,23,31,22,19)(18,27,39,24,26,44,28,38)(20,36,45,34,35,47,37,30)(25,42,48,40,33,46,43,41)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=ghi,d^2=gi,e^2=fg,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, 7515335);