#409715

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

912 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316911);