#7181067

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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

1685 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)(23,34)(24,30)(25,33)(26,32)(27,31)(28,37)(29,39)(35,48)(36,45)(38,44)(40,49)(41,50)(42,47)(43,51)(46,52);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,37)(22,28,32,34)(24,38,40,35)(25,41,27,36)(29,46,47,43)(30,48,49,44)(31,50,33,45)(39,51,42,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,26,40)(22,29,32,47)(23,35,37,38)(25,39,27,42)(28,43,34,46)(30,33,49,31)(36,51,41,52)(44,45,48,50);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,24)(22,30)(23,35)(25,42)(26,40)(27,39)(28,44)(29,31)(32,49)(33,47)(34,48)(36,52)(37,38)(41,51)(43,50)(45,46);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,25,26,27)(22,31,32,33)(23,36,37,41)(24,39,40,42)(28,45,34,50)(29,30,47,49)(35,51,38,52)(43,48,46,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=i,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7088263);