#483482

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

406 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)(17,18,21,26,23,27,22,37)(19,28,32,45,30,43,33,48)(20,34,29,42,36,39,35,25)(24,38,41,46,40,47,44,31);(17,19,29,38,22,33,20,31,23,30,35,47,21,32,36,46)(18,24,39,48,26,41,25,28,27,40,34,45,37,44,42,43);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(19,30)(24,40)(28,43)(31,46)(32,33)(38,47)(41,44)(45,48);(3,9)(6,12)(10,15)(13,16)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,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	#72320258, #72321746, #72321901
maximal normal subgroup of	#72320258, #72321746, #72321901

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390678);