#495352

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
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

808 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)(9,10,15,21,16,22,36,49,14,20,37,50,24,39,13,19)(11,23,28,41,29,48,26,51,27,53,35,17,38,42,60,43)(12,30,33,63,34,64,65,45,31,18,44,46,54,47,32,62)(25,55,58,40,59,68,57,69,56,52,66,61,67,71,70,72);(2,5)(6,7)(9,11,24,29,14,27,16,38)(10,17,39,43,20,41,22,51)(12,25,54,59,31,56,34,67)(13,35,37,60,36,28,15,26)(18,40,64,69,30,61,47,72)(19,48,50,53,49,42,21,23)(32,66,44,70,65,58,33,57)(45,71,63,55,62,68,46,52);(1,3,4,8)(2,6,5,7)(9,12,14,31)(10,18,20,30)(11,25,27,56)(13,32,36,65)(15,33,37,44)(16,34,24,54)(17,40,41,61)(19,45,49,62)(21,46,50,63)(22,47,39,64)(23,52,53,55)(26,57,60,70)(28,58,35,66)(29,59,38,67)(42,68,48,71)(43,69,51,72);(12,31)(18,30)(25,56)(32,65)(33,44)(34,54)(40,61)(45,62)(46,63)(47,64)(52,55)(57,70)(58,66)(59,67)(68,71)(69,72)

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=f,e^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,e^b=eh,g^b=gh,h^b=hi,d^c=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, 402548);