#493124

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂⁴	#131
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

760 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=i,f^2=h,h^2=i,b^a=be,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400320);