#7608176

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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

1256 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)(19,29)(20,32)(21,24)(22,38)(23,26)(25,34)(27,31)(28,39)(30,43)(33,45)(35,42)(36,44)(37,41)(40,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,22,21,23,31,39,37)(18,24,28,29,26,41,38,27)(20,33,36,48,34,30,46,35)(25,42,44,43,32,40,47,45);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,20)(18,25)(19,30)(21,35)(22,36)(23,34)(24,40)(26,32)(27,43)(28,44)(29,45)(31,33)(37,48)(38,47)(39,46)(41,42);(3,8)(6,7)(9,12,14,13)(10,11,15,16)(18,26)(19,31)(20,34)(21,37)(28,38)(36,46)(40,42)(43,45);(9,13,14,12)(10,11,15,16)(17,21,22,31,23,37,39,19)(18,27,28,24,26,29,38,41)(20,35,36,33,34,48,46,30)(25,43,44,40,32,45,47,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gh,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,e^d=ei

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, 7515372);