#7199913

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,5,7)(3,6,4,8)(10,14)(11,17)(12,20)(22,24)(25,26,31,37,32,38,30,36)(27,34,40,50,28,41,43,33)(29,44,47,53,48,35,46,54)(39,52,45,49,42,51,55,56);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,18,17,15)(12,21,20,16)(19,22,23,24)(25,27,32,28)(26,33,38,50)(29,45,48,55)(30,40,31,43)(34,37,41,36)(35,51,44,52)(39,46,42,47)(49,53,56,54);(1,4,5,3)(2,6,7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,32,27)(26,34,38,41)(29,39,48,42)(30,40,31,43)(33,37,50,36)(35,52,44,51)(45,46,55,47)(49,53,56,54);(10,14)(11,17)(16,21)(19,23)(25,29,30,46,32,48,31,47)(26,35,36,53,38,44,37,54)(27,39,43,55,28,42,40,45)(33,49,41,52,50,56,34,51);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(25,30,32,31)(26,36,38,37)(27,40,28,43)(29,46,48,47)(33,34,50,41)(35,53,44,54)(39,45,42,55)(49,51,56,52)

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=g,d^2=fhi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7107109);