#493153

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	C₂×C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

720 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,e^2=hi,f^2=h,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, 400349);