#483869

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	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fi,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=dhi,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, 391065);