#7186781

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 7093977);