#7104653

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1674 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)(21,22,28,41,27,25,26,31)(23,32,42,29,34,47,35,39)(24,30,43,52,37,40,38,46)(33,48,51,44,36,49,50,45);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,34)(22,29,25,39)(24,36,37,33)(26,42,28,35)(30,45,40,44)(31,47,41,32)(38,50,43,51)(46,48,52,49);(2,4)(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24,27,37)(22,30,25,40)(23,33,34,36)(26,38,28,43)(29,44,39,45)(31,46,41,52)(32,48,47,49)(35,50,42,51);(8,16)(11,20)(13,15)(17,19)(21,25,28,31,27,22,26,41)(23,32,42,29,34,47,35,39)(24,30,43,52,37,40,38,46)(33,49,51,45,36,48,50,44);(6,12)(7,14)(11,20)(13,15)(22,25)(23,34)(24,37)(31,41)(35,42)(38,43)(44,45)(48,49)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=h,d^2=fi,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, 7011849);