#483434

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
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	C₂²×#107
prime radical	C₂⁴	#107
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

452 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ch,d^b=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, 390630);