#7199969

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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

1250 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)(25,26,30,35,31,34,45,52)(27,36,39,48,38,53,29,32)(28,40,44,56,43,33,49,51)(37,55,42,47,41,54,46,50);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,18,17,15)(12,21,20,16)(19,22,23,24)(25,27,31,38)(26,32,34,48)(28,41,43,37)(29,45,39,30)(33,50,40,47)(35,53,52,36)(42,49,46,44)(51,54,56,55);(1,4,5,3)(2,6,7,8)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,28,31,43)(26,33,34,40)(27,37,38,41)(29,46,39,42)(30,44,45,49)(32,47,48,50)(35,51,52,56)(36,54,53,55);(10,14)(11,17)(16,21)(19,23)(25,29,30,27,31,39,45,38)(26,32,35,36,34,48,52,53)(28,42,44,41,43,46,49,37)(33,50,51,55,40,47,56,54);(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(26,34)(27,38)(28,43)(29,39)(35,52)(44,49)(47,50)(54,55)

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,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7107165);