#7361116

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	#42995	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 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)(3,7)(4,6)(5,8)(17,18,23,27)(19,24,22,35)(20,30,32,37)(21,26,34,42)(25,36,40,28)(29,38,33,41)(31,44,46,47)(39,45,48,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,35)(20,28,32,36)(21,33,34,29)(25,37,40,30)(26,41,42,38)(31,45,46,43)(39,44,48,47);(9,11,12,16)(10,13,14,15)(17,20,23,32)(18,25,27,40)(19,28,22,36)(21,31,34,46)(24,37,35,30)(26,39,42,48)(29,43,33,45)(38,47,41,44);(3,8)(5,7)(10,14)(11,16)(17,21)(18,26)(19,29)(20,31)(22,33)(23,34)(24,38)(25,39)(27,42)(28,43)(30,44)(32,46)(35,41)(36,45)(37,47)(40,48);(2,6)(5,7)(10,14)(13,15)(20,32)(21,34)(25,40)(26,42)(28,36)(29,33)(30,37)(38,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,e^c=ei,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, 7268312);