#7586226

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(10,14)(11,12)(17,18,22,28)(19,29,33,24)(20,26,36,25)(21,37,23,27)(30,40,34,44)(31,47,48,41)(32,42,39,43)(35,45,38,46);(2,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,33)(18,24,28,29)(20,34,36,30)(21,32,23,39)(25,44,26,40)(27,42,37,43)(31,38,48,35)(41,46,47,45);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,38)(22,36)(23,35)(24,40)(26,28)(27,46)(29,44)(31,32)(33,34)(37,45)(39,48)(41,42)(43,47);(9,11,13,12)(10,16,14,15)(17,20)(18,26)(19,31)(21,35)(22,36)(23,38)(24,41)(25,28)(27,46)(29,47)(30,32)(33,48)(34,39)(37,45)(40,42)(43,44);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,28,37)(19,32,33,39)(20,35,36,38)(24,42,29,43)(25,45,26,46)(30,48,34,31)(40,47,44,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=gi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,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, 7493422);