#7211480

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

1230 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)(25,26,30,35)(27,31,38,34)(28,39,42,46)(29,43,45,33)(32,40,49,36)(37,50,44,47)(41,54,55,52)(48,51,56,53);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,38)(26,31,35,34)(28,40,42,36)(29,44,45,37)(32,39,49,46)(33,50,43,47)(41,51,55,53)(48,54,56,52);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,38,40,30,42,27,36)(26,32,31,46,35,49,34,39)(29,41,44,53,45,55,37,51)(33,48,47,54,43,56,50,52);(2,6)(3,7)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,29,30,45)(26,33,35,43)(27,37,38,44)(28,41,42,55)(31,47,34,50)(32,48,49,56)(36,51,40,53)(39,52,46,54);(3,7)(5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(29,45)(33,43)(37,44)(41,55)(47,50)(48,56)(51,53)(52,54)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,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^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, 7118676);