#7205645

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	#30398	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1479 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)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,24,29)(19,25,34,42)(20,35,21,27)(22,28,23,38)(26,37,43,36)(30,44,31,40)(32,41,33,47)(39,46,48,45);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,24,34)(18,25,29,42)(20,30,21,31)(22,32,23,33)(26,39,43,48)(27,40,35,44)(28,41,38,47)(36,45,37,46);(3,7)(5,8)(10,14)(15,16)(17,20,24,21)(18,26,29,43)(19,30,34,31)(22,36,23,37)(25,39,42,48)(27,38,35,28)(32,45,33,46)(40,47,44,41);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,21,24,20)(18,27,29,35)(19,31,34,30)(22,36,23,37)(25,40,42,44)(26,38,43,28)(32,45,33,46)(39,47,48,41);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,22,24,23)(18,28,29,38)(19,32,34,33)(20,36,21,37)(25,41,42,47)(26,27,43,35)(30,45,31,46)(39,40,48,44)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7112841);