#7183973

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 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)(8,16)(11,17)(15,20)(18,19)(21,22,25,30,26,31,38,27)(23,28,34,43,33,42,48,39)(24,29,35,46,37,45,50,40)(32,41,47,52,36,44,49,51);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(22,27)(24,35)(25,38)(28,39)(30,31)(32,47)(34,48)(36,49)(37,50)(40,46)(42,43)(51,52);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,23,26,33)(22,28,31,42)(24,36,37,32)(25,34,38,48)(27,39,30,43)(29,44,45,41)(35,49,50,47)(40,52,46,51);(7,14)(10,13)(15,20)(18,19)(21,24,25,35,26,37,38,50)(22,29,30,46,31,45,27,40)(23,32,34,47,33,36,48,49)(28,41,43,52,42,44,39,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(23,33)(24,37)(28,42)(29,45)(34,48)(35,50)(39,43)(40,46)

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=i,d^2=f,e^2=h,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=dhi,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, 7091169);