#7181147

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7088343);