#7167138

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 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)(3,4)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(22,27)(25,28)(29,41)(30,36)(31,42)(32,48)(35,43)(38,40)(44,52)(45,49)(46,47)(50,51);(7,13)(8,16)(10,17)(11,15)(21,22,28,42,26,31,25,27)(23,32,43,41,34,29,35,48)(24,36,40,46,39,47,38,30)(33,49,51,44,37,52,50,45);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,29)(24,37)(25,35)(26,34)(27,41)(28,43)(30,45)(31,32)(33,39)(36,49)(38,51)(40,50)(42,48)(44,46)(47,52);(7,13)(10,17)(14,19)(18,20)(21,24,26,39)(22,30,31,46)(23,33,34,37)(25,40,28,38)(27,36,42,47)(29,44,32,45)(35,51,43,50)(41,52,48,49);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,31)(23,34)(27,42)(30,46)(33,37)(35,43)(36,47)(50,51)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

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, 7074334);