#7199880

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=gi,d^2=fh,e^2=hi,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

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, 7107076);