#7536043

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

1626 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(18,23)(19,27)(20,31)(26,35)(30,42)(33,36)(38,43)(45,48);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,23)(19,28,27,24)(20,32,31,40)(22,35,34,26)(25,36,37,33)(29,45,41,48)(30,39,42,44)(38,46,43,47);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19,21,27)(18,24,23,28)(20,29,31,41)(22,30,34,42)(25,38,37,43)(26,39,35,44)(32,48,40,45)(33,46,36,47);(10,14)(13,15)(17,20,21,31)(18,25,23,37)(19,29,27,41)(22,36,34,33)(24,38,28,43)(26,40,35,32)(30,47,42,46)(39,45,44,48);(2,5)(6,8)(10,14)(13,15)(19,27)(24,28)(29,41)(30,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fgi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,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, 7443239);