#7201480

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7108676);