#7532354

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	#45241	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 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)(17,18,22,27)(19,28,32,24)(20,25,34,43)(21,35,23,26)(29,45,47,38)(30,46,48,39)(31,40,37,41)(33,44,36,42);(2,5)(6,8)(11,16)(13,15)(17,19)(18,24)(20,29)(21,31)(22,32)(23,37)(25,38)(26,40)(27,28)(30,33)(34,47)(35,41)(36,48)(39,44)(42,46)(43,45);(9,11,12,16)(10,13,14,15)(17,20,22,34)(18,25,27,43)(19,29,32,47)(21,36,23,33)(24,38,28,45)(26,44,35,42)(30,31,48,37)(39,41,46,40);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,33)(22,34)(23,36)(24,39)(26,42)(27,43)(28,46)(29,37)(31,47)(32,48)(35,44)(38,40)(41,45);(2,5)(6,8)(17,21,22,23)(18,26,27,35)(19,31,32,37)(20,33,34,36)(24,40,28,41)(25,42,43,44)(29,30,47,48)(38,46,45,39)

Polycyclic presentation:

2^9|a^2=i,c^2=gi,d^2=fi,e^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,d^c=di,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, 7439550);