#7539993

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	#36011	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1551 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)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,32,41)(20,25,21,26)(22,27,37,46)(29,38,34,43)(30,39,47,48)(31,40,36,45)(33,42,35,44);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,23,32)(18,24,28,41)(20,29,21,34)(22,36,37,31)(25,38,26,43)(27,45,46,40)(30,33,47,35)(39,42,48,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,23,21)(18,25,28,26)(19,29,32,34)(22,33,37,35)(24,38,41,43)(27,42,46,44)(30,36,47,31)(39,45,48,40);(10,14)(13,15)(17,21,23,20)(18,26,28,25)(19,30,32,47)(22,33,37,35)(24,39,41,48)(27,42,46,44)(29,36,34,31)(38,45,43,40);(2,5)(6,8)(10,14)(13,15)(20,21)(25,26)(29,34)(30,47)(33,35)(38,43)(39,48)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7447189);