#353247

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₄	C₂×C₄³
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1378 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=ef,e^2=i,b^a=be,c^a=ci,d^a=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260443);