#6792306

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

1348 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)(27,33)(28,43)(29,35)(30,37)(31,36)(32,38)(34,44)(39,53)(40,48)(41,54)(42,50)(45,52)(46,49)(47,56)(51,55);(1,2)(3,7)(4,5)(6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,32,42)(26,33,38,50)(28,40,29,39)(30,46,31,41)(34,51,52,47)(35,53,43,48)(36,54,37,49)(44,55,45,56);(2,5)(6,7)(25,28,32,29)(26,34,38,52)(27,39,42,40)(30,44,31,45)(33,47,50,51)(35,36,43,37)(41,55,46,56)(48,54,53,49);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,32,28)(26,35,38,43)(27,40,42,39)(30,44,31,45)(33,48,50,53)(34,36,52,37)(41,55,46,56)(47,54,51,49);(1,3,4,8)(2,6,5,7)(11,19)(14,17)(18,24)(22,23)(25,30,32,31)(26,36,38,37)(27,41,42,46)(28,44,29,45)(33,49,50,54)(34,43,52,35)(39,55,40,56)(47,48,51,53)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6699502);