#7574886

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

1312 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)(9,10)(11,14)(12,13)(15,16)(17,18,23,29)(19,25,34,42)(20,27,21,26)(22,28,24,30)(31,43,35,39)(32,48,47,40)(33,41,38,46)(36,45,37,44);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,35)(21,31)(22,33)(23,34)(24,38)(26,43)(27,39)(28,41)(29,42)(30,46)(32,37)(36,47)(40,45)(44,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,26)(19,31)(21,23)(22,37)(24,36)(25,39)(27,29)(28,45)(30,44)(32,38)(33,47)(34,35)(40,46)(41,48)(42,43);(10,13)(14,16)(17,21,23,20)(18,27,29,26)(19,32,34,47)(22,37,24,36)(25,40,42,48)(28,45,30,44)(31,33,35,38)(39,41,43,46);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(17,22,23,24)(18,28,29,30)(19,33,34,38)(20,36,21,37)(25,41,42,46)(26,44,27,45)(31,32,35,47)(39,40,43,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=g,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, 7482082);