#7024270

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1517 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)(17,18,22,23)(19,28,31,41)(20,25,32,38)(21,26,36,27)(24,39,37,30)(29,45,33,48)(34,43,35,44)(40,46,42,47);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(18,23)(20,32)(24,37)(26,27)(28,41)(29,33)(34,35)(46,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,30,22,31,36,39)(18,24,27,28,23,37,26,41)(20,33,35,46,32,29,34,47)(25,42,43,48,38,40,44,45);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,35)(22,32)(23,38)(24,40)(26,44)(27,43)(28,45)(30,47)(31,33)(34,36)(37,42)(39,46)(41,48);(10,14)(13,15)(17,21,22,36)(18,26,23,27)(19,30,31,39)(20,34,32,35)(24,41,37,28)(25,43,38,44)(29,46,33,47)(40,45,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fgi,d^2=fi,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,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6931466);