#371417

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1071 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278613);