#7032580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1368 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6939776);