#6793007

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1320 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)(11,19)(12,16)(13,18)(14,17)(15,21)(20,24)(22,23)(25,26,29,34)(27,36,38,31)(28,40,42,33)(30,32,44,35)(37,46,41,49)(39,47,52,45)(43,50,54,48)(51,55,53,56);(1,2)(3,7)(4,5)(6,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27,29,38)(26,31,34,36)(28,41,42,37)(30,39,44,52)(32,45,35,47)(33,49,40,46)(43,53,54,51)(48,56,50,55);(2,5)(6,7)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(26,32)(30,44)(31,45)(33,48)(34,35)(36,47)(39,52)(40,50)(43,54)(46,55)(49,56)(51,53);(10,17)(15,19)(16,18)(23,24)(25,28,29,42)(26,33,34,40)(27,37,38,41)(30,43,44,54)(31,46,36,49)(32,48,35,50)(39,51,52,53)(45,55,47,56);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,18,17,16)(11,20,21,22)(15,23,19,24)(26,34)(27,38)(28,42)(32,35)(39,52)(43,54)(46,49)(55,56)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=i,e^2=fhi,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, 6700203);