#6648928

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632082
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1221 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=i,d^2=h,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=dfhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6556124);