#7211349

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1228 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,e^c=ei,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, 7118545);