#7034380

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1572 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)(3,6)(4,5)(7,8)(9,10,12,13)(11,14,16,15)(17,18)(19,24)(20,26)(21,27)(22,36)(23,28)(25,33)(29,38)(30,39)(31,46)(32,40)(34,41)(35,42)(37,43)(44,47)(45,48);(2,5)(3,7)(9,10)(11,15)(12,13)(14,16)(17,19,23,32)(18,24,28,40)(20,29,34,44)(21,30,22,31)(25,37,42,48)(26,38,41,47)(27,39,36,46)(33,43,35,45);(10,13)(11,16)(17,20,23,34)(18,25,28,42)(19,29,32,44)(21,33,22,35)(24,37,40,48)(26,27,41,36)(30,43,31,45)(38,39,47,46);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,20)(18,26)(19,29)(21,35)(22,33)(23,34)(24,38)(25,36)(27,42)(28,41)(30,45)(31,43)(32,44)(37,46)(39,48)(40,47);(10,13)(14,15)(17,21,23,22)(18,27,28,36)(19,30,32,31)(20,33,34,35)(24,39,40,46)(25,41,42,26)(29,43,44,45)(37,47,48,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6941576);