#7556619

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	#36015	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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)(4,5)(6,8)(9,10,13,14)(11,16,12,15)(17,18,22,26)(19,23,30,38)(20,24,32,40)(21,25,34,42)(27,35,31,39)(28,36,29,37)(33,41,45,48)(43,46,44,47);(2,5)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,30)(18,23,26,38)(20,31,32,27)(21,28,34,29)(24,39,40,35)(25,36,42,37)(33,44,45,43)(41,47,48,46);(10,14)(15,16)(17,20)(18,24)(19,27)(21,33)(22,32)(23,35)(25,41)(26,40)(28,43)(29,44)(30,31)(34,45)(36,46)(37,47)(38,39)(42,48);(9,11,13,12)(10,15,14,16)(19,28)(21,34)(23,36)(25,42)(27,43)(29,30)(31,44)(33,45)(35,46)(37,38)(39,47)(41,48);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(20,32)(24,40)(27,31)(33,45)(35,39)(41,48)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,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, 7463815);