#483481

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
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	C₂×#41	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

405 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,d^c=dhi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260370, #72260372, #72260411
maximal normal subgroup of	#72260370, #72260372, #72260411

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390677);