#7183943

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=i,d^2=fi,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=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, 7091139);