#493156

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#107
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400352);