#7182615

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

1659 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fhi,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,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, 7089811);