#7024078

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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)(20,25,33,38)(21,26,36,27)(24,30,37,32)(29,43,44,48)(34,40,35,42)(39,46,47,45);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(18,23)(20,33)(24,37)(26,27)(28,41)(29,44)(34,35)(45,46);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,32,22,31,36,30)(18,24,27,28,23,37,26,41)(20,29,35,45,33,44,34,46)(25,39,40,43,38,47,42,48);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,20)(18,25)(19,29)(21,35)(22,33)(23,38)(24,39)(26,42)(27,40)(28,43)(30,46)(31,44)(32,45)(34,36)(37,47)(41,48);(10,14)(13,15)(17,21,22,36)(18,26,23,27)(19,30,31,32)(20,34,33,35)(24,28,37,41)(25,40,38,42)(29,45,44,46)(39,48,47,43)

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=dh,e^c=ei,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, 6931274);