#494091

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×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

624 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)(3,6)(4,5)(7,8)(9,10,15,21,16,23,35,22)(11,24,29,52,26,51,59,40)(12,30,33,44,34,62,63,61)(13,20,14,19,36,48,37,49)(17,28,41,57,39,58,42,27)(18,43,45,64,47,31,46,32)(25,38,56,66,53,65,72,67)(50,68,70,71,69,54,60,55);(2,5)(6,8)(9,11,14,28,15,29,36,57,16,26,37,58,35,59,13,27)(10,17,20,40,22,42,49,51,23,39,48,52,21,41,19,24)(12,25,32,55,33,56,43,68,34,53,64,71,63,72,31,54)(18,38,30,60,46,67,61,69,47,65,62,70,45,66,44,50);(1,3,4,7)(2,6,5,8)(9,12,13,31,35,63,37,64,16,34,36,43,15,33,14,32)(10,18,19,44,21,45,48,62,23,47,49,61,22,46,20,30)(11,25,27,54,59,72,58,71,26,53,57,68,29,56,28,55)(17,38,24,50,41,66,52,70,39,65,51,69,42,67,40,60);(11,26)(17,39)(24,51)(25,53)(27,57)(28,58)(29,59)(38,65)(40,52)(41,42)(50,69)(54,68)(55,71)(56,72)(60,70)(66,67)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^b=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, 401287);