#495382

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,7)(4,5)(6,8)(9,10,15,20,16,21,14,37)(11,22,26,47,27,48,25,56)(12,28,32,43,29,58,31,18)(13,33,35,39,36,64,34,19)(17,38,41,55,42,54,46,24)(23,49,53,67,50,69,52,45)(30,61,63,72,60,44,62,59)(40,51,57,65,66,70,68,71);(2,5)(6,7)(9,11,13,24,15,26,35,38,16,27,36,55,14,25,34,54)(10,17,39,48,37,46,33,47,21,42,19,22,20,41,64,56)(12,23,30,51,32,53,63,65,29,50,60,70,31,52,62,71)(18,40,61,67,58,68,59,49,43,66,44,45,28,57,72,69);(1,3,4,8)(2,6,5,7)(9,12,14,31,16,29,15,32)(10,18,20,28,21,43,37,58)(11,23,25,52,27,50,26,53)(13,30,34,62,36,60,35,63)(17,40,41,57,42,66,46,68)(19,44,33,59,39,61,64,72)(22,45,47,49,48,67,56,69)(24,51,54,71,55,70,38,65);(12,29)(18,43)(23,50)(28,58)(30,60)(31,32)(40,66)(44,61)(45,67)(49,69)(51,70)(52,53)(57,68)(59,72)(62,63)(65,71)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402578);