#7301508

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	#42992	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1909 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7208704);