#7608214

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

1274 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)(17,18,23,28)(19,26,32,44)(20,25,33,40)(21,36,31,24)(22,38,29,27)(30,43,46,41)(34,39,37,48)(35,47,45,42);(1,3)(2,6)(4,7)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19,29,21,23,32,22,31)(18,24,38,44,28,36,27,26)(20,30,45,37,33,46,35,34)(25,39,47,41,40,48,42,43);(1,4,5,2)(3,7,8,6)(10,15)(11,16)(17,20)(18,25)(19,30)(21,37)(22,35)(23,33)(24,39)(26,43)(27,42)(28,40)(29,45)(31,34)(32,46)(36,48)(38,47)(41,44);(3,8)(6,7)(9,12,14,13)(10,11,15,16)(20,33)(25,40)(30,46)(34,37)(35,45)(39,48)(41,43)(42,47);(9,13,14,12)(10,11,15,16)(17,21,29,32,23,31,22,19)(18,26,38,24,28,44,27,36)(20,34,45,30,33,37,35,46)(25,41,47,48,40,43,42,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=gh,d^2=g,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

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, 7515410);