#483872

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	#2186	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

434 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=i,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.

Relationships

maximal subgroup of	#72260325, #72260361, #72260370, …
maximal normal subgroup of	#72260325, #72260361, #72260370, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391068);