#483791

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fi,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=di

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, 390987);