#7528393

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(17,18,23,28)(19,24,20,25)(21,26,35,44)(22,27,37,46)(29,38,31,40)(30,39,32,41)(33,42,47,48)(34,43,36,45);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,20)(18,24,28,25)(21,33,35,47)(22,32,37,30)(26,42,44,48)(27,41,46,39)(29,36,31,34)(38,45,40,43);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,23,19)(18,25,28,24)(21,31,35,29)(22,32,37,30)(26,40,44,38)(27,41,46,39)(33,34,47,36)(42,43,48,45);(10,14)(13,16)(17,21,23,35)(18,26,28,44)(19,29,20,31)(22,36,37,34)(24,38,25,40)(27,45,46,43)(30,47,32,33)(39,48,41,42);(1,4,5,3)(2,7,6,8)(19,20)(24,25)(29,31)(30,32)(33,47)(38,40)(39,41)(42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fi,d^2=i,e^2=f,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, 7435589);