#6913254

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1385 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)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,29,31)(27,35,38,50)(28,40,41,33)(30,44,36,34)(32,46,45,39)(37,53,42,51)(43,49,52,55)(47,54,48,56);(1,2)(3,7)(4,6)(5,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(26,31)(28,41)(32,45)(34,44)(35,50)(37,42)(43,52)(54,56);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15)(17,19)(18,22)(20,24)(21,23)(25,27,36,46,29,38,30,39)(26,32,44,50,31,45,34,35)(28,42,52,54,41,37,43,56)(33,48,55,51,40,47,49,53);(3,8)(5,7)(10,15)(14,17)(16,23)(22,24)(25,28)(26,33)(27,37)(29,41)(30,43)(31,40)(32,47)(34,49)(35,51)(36,52)(38,42)(39,54)(44,55)(45,48)(46,56)(50,53);(2,6)(5,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(28,41)(33,40)(37,42)(43,52)(47,48)(49,55)(51,53)(54,56)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,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,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6820450);