#6937127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884841
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1282 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,c^2=fi,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh,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, 6844323);