#7528390

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=f,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7435586);