#7587018

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1495 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)(3,8)(4,6)(5,7)(10,14)(11,12)(17,18,22,28)(19,29,33,24)(20,26,35,25)(21,36,23,27)(30,39,47,45)(31,46,48,40)(32,41,38,42)(34,43,37,44);(1,3,5,4)(2,6,7,8)(9,10,13,14)(11,16,12,15)(17,19,22,33)(18,24,28,29)(20,30,35,47)(21,32,23,38)(25,39,26,45)(27,41,36,42)(31,37,48,34)(40,44,46,43);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,30)(21,37)(22,35)(23,34)(24,39)(26,28)(27,44)(29,45)(31,38)(32,48)(33,47)(36,43)(40,42)(41,46);(9,11,13,12)(10,16,14,15)(17,20)(18,26)(19,31)(21,34)(22,35)(23,37)(24,40)(25,28)(27,44)(29,46)(30,38)(32,47)(33,48)(36,43)(39,42)(41,45);(2,7)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,28,36)(19,32,33,38)(20,34,35,37)(24,41,29,42)(25,43,26,44)(30,31,47,48)(39,40,45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=f,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=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7494214);