#6775795

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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

1203 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=hi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6682991);