#6777590

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

Polycyclic presentation:

2^9|a^2=fi,c^2=f,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,e^c=ei,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, 6684786);