#7034878

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1784 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)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,28,31,27,25,26,32)(23,33,43,29,36,45,35,41)(24,30,38,48,40,42,39,47)(34,49,51,44,37,50,52,46);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,36)(22,29,25,41)(24,37,40,34)(26,43,28,35)(30,46,42,44)(31,33,32,45)(38,51,39,52)(47,49,48,50);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24)(22,30)(23,34)(25,42)(26,39)(27,40)(28,38)(29,44)(31,48)(32,47)(33,49)(35,52)(36,37)(41,46)(43,51)(45,50);(8,16)(11,20)(13,15)(17,19)(21,25,28,32,27,22,26,31)(23,33,43,29,36,45,35,41)(24,30,38,48,40,42,39,47)(34,50,51,46,37,49,52,44);(6,12)(7,14)(11,20)(13,15)(21,26,27,28)(22,31,25,32)(23,35,36,43)(24,38,40,39)(29,45,41,33)(30,47,42,48)(34,51,37,52)(44,49,46,50)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6942074);