#7181097

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1678 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7088293);