#6640528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636940
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 ◁ #2293 ◁ 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)(9,10,14,17)(11,18,20,15)(12,16,13,22)(19,24,21,23)(25,26,30,34)(27,31,36,45)(28,38,41,55)(29,43,39,33)(32,47,49,42)(35,50,40,54)(37,53,51,46)(44,56,48,52);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,23)(17,18)(22,24)(25,27)(26,31)(28,35)(29,37)(30,36)(32,44)(33,46)(34,45)(38,50)(39,51)(40,41)(42,52)(43,53)(47,56)(48,49)(54,55);(3,7)(5,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28,39,47,30,41,29,42)(26,32,43,55,34,49,33,38)(27,35,51,56,36,40,37,52)(31,44,53,54,45,48,46,50);(3,7)(5,8)(10,17)(11,20)(16,22)(19,21)(25,27)(26,31)(28,40)(29,37)(30,36)(32,48)(33,46)(34,45)(35,41)(38,54)(39,51)(42,56)(43,53)(44,49)(47,52)(50,55);(2,6)(5,8)(9,13,14,12)(10,16,17,22)(11,21,20,19)(15,23,18,24)(27,36)(28,41)(31,45)(32,49)(37,51)(38,55)(42,47)(46,53)

Polycyclic presentation:

2^9|a^2=fi,c^2=fgi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,e^b=ei,d^c=di,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, 6547724);