#6776514

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=i,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6683710);