#6792153

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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1358 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6699349);