#7186575

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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

1676 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)(23,28)(24,35)(25,30)(26,31)(27,32)(29,36)(33,44)(34,41)(37,48)(38,42)(39,46)(40,45)(43,49)(47,52)(50,51);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,26)(22,28,32,31)(24,36,38,33)(25,34,40,39)(29,42,44,35)(30,41,45,46)(37,49,50,47)(43,51,52,48);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,26,36,27,38,23,33)(22,29,28,35,32,44,31,42)(25,37,39,49,40,50,34,47)(30,43,41,48,45,52,46,51);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,25,27,40)(22,30,32,45)(23,34,26,39)(24,37,38,50)(28,41,31,46)(29,43,44,52)(33,47,36,49)(35,48,42,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,23,27,26)(22,31,32,28)(24,33,38,36)(25,39,40,34)(29,42,44,35)(30,41,45,46)(37,49,50,47)(43,48,52,51)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,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=ehi,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7093771);