#6777645

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1233 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6684841);