#7015620

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1284 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)(17,18,22,27,23,28,21,35)(19,24,30,41,31,42,36,40)(20,26,33,45,34,44,37,25)(29,39,48,43,32,46,47,38);(1,2,5,6)(3,8,4,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,31)(18,24,28,42)(20,32,34,29)(21,36,22,30)(25,43,45,38)(26,46,44,39)(27,41,35,40)(33,47,37,48);(9,11)(10,13)(12,16)(14,15)(17,20,23,34)(18,25,28,45)(19,29,31,32)(21,33,22,37)(24,38,42,43)(26,27,44,35)(30,47,36,48)(39,41,46,40);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20)(18,26)(19,29)(21,37)(22,33)(23,34)(24,39)(25,35)(27,45)(28,44)(30,48)(31,32)(36,47)(38,40)(41,43)(42,46);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,23,22)(18,27,28,35)(19,30,31,36)(20,33,34,37)(24,40,42,41)(25,44,45,26)(29,47,32,48)(38,39,43,46)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 6922816);