#494797

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂×C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

582 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)(9,10,14,21,16,19,15,36)(11,17,25,40,23,41,26,56)(12,27,29,59,30,58,37,46)(13,31,33,45,35,61,34,20)(18,32,43,60,42,62,44,28)(22,47,50,68,48,69,57,66)(24,52,54,67,51,71,55,39)(38,53,63,70,65,72,64,49);(2,5)(6,8)(9,11,14,25,16,23,15,26)(10,17,21,40,19,41,36,56)(12,22,29,50,30,48,37,57)(13,24,33,54,35,51,34,55)(18,38,43,63,42,65,44,64)(20,39,31,52,45,67,61,71)(27,47,59,68,58,69,46,66)(28,49,32,53,60,70,62,72);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,32)(14,29)(15,37)(16,30)(17,38)(19,42)(20,46)(21,43)(23,48)(24,53)(25,50)(26,57)(27,31)(28,34)(33,60)(35,62)(36,44)(39,66)(40,63)(41,65)(45,59)(47,52)(49,55)(51,72)(54,70)(56,64)(58,61)(67,68)(69,71);(10,19)(11,23)(18,42)(20,45)(21,36)(22,48)(24,51)(25,26)(27,58)(31,61)(43,44)(46,59)(49,70)(50,57)(53,72)(54,55)

Polycyclic presentation:

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

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, 401993);