#7200901

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

1420 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,35)(27,37,40,32)(28,34,42,50)(29,36,31,33)(38,51,53,46)(39,47,43,45)(41,48,44,49)(52,55,54,56);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27)(26,32)(28,38)(29,43)(30,40)(31,39)(33,45)(34,46)(35,37)(36,47)(41,54)(42,53)(44,52)(48,55)(49,56)(50,51);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(29,31)(32,45)(34,48)(35,36)(37,47)(39,43)(41,44)(46,55)(49,50)(51,56)(52,54);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,28,30,42)(26,34,35,50)(27,38,40,53)(29,44,31,41)(32,46,37,51)(33,48,36,49)(39,54,43,52)(45,55,47,56);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(25,29,30,31)(26,33,35,36)(27,39,40,43)(28,41,42,44)(32,47,37,45)(34,49,50,48)(38,52,53,54)(46,55,51,56)

Polycyclic presentation:

2^9|a^2=i,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,e^c=ei,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, 7108097);