#7532259

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	#45134	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=f,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, 7439455);