#7608240

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 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)(22,39)(23,26)(25,34)(27,31)(28,40)(30,46)(33,44)(35,41)(36,45)(37,43)(38,42)(47,48);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,22,21,23,31,40,38)(18,24,28,29,26,42,39,27)(20,33,36,35,34,30,47,37)(25,43,45,46,32,41,48,44);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,20)(18,25)(19,30)(21,37)(22,36)(23,34)(24,41)(26,32)(27,46)(28,45)(29,44)(31,33)(35,38)(39,48)(40,47)(42,43);(3,8)(6,7)(9,12,14,13)(10,11,15,16)(18,26)(19,31)(20,34)(21,38)(28,39)(36,47)(41,43)(44,46);(9,13,14,12)(10,11,15,16)(17,21,22,31,23,38,40,19)(18,27,28,24,26,29,39,42)(20,35,36,30,34,37,47,33)(25,44,45,43,32,46,48,41)

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^c=ei,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, 7515436);