#493152

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

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,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, 400348);