#7035311

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

1226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(2,5)(6,7)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,35)(28,33)(29,31)(30,42)(32,38)(34,41)(36,48)(37,49)(39,45)(40,53)(43,44)(46,51)(47,52)(50,56)(54,55);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(26,31)(27,36)(28,39)(34,42)(38,43)(40,52)(46,55)(49,56);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,27,29,36)(26,32,31,44)(28,37,39,50)(30,43,41,38)(33,46,45,55)(34,48,42,35)(40,54,52,51)(47,56,53,49);(1,3,4,8)(2,6,5,7)(12,20)(15,24)(17,19)(21,23)(25,28,29,39)(26,33,31,45)(27,37,36,50)(30,40,41,52)(32,46,44,55)(34,47,42,53)(35,49,48,56)(38,51,43,54);(10,16)(11,18)(15,24)(17,19)(26,31)(28,39)(32,44)(34,42)(35,48)(37,50)(40,52)(51,54)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=f,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,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, 6942507);