#492975

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×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	C₂²×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1072 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)(3,4)(5,6,10,16,11,15,30,39,12,17,31,40,9,28,29,54)(7,18,22,43,23,34,49,58,20,42,48,63,21,13,32,35)(8,14,26,38,27,37,53,60,24,36,50,59,25,51,52,66)(19,41,45,61,46,57,64,67,47,62,65,68,44,33,55,56);(2,4)(5,7)(6,13,17,34)(8,19)(9,21)(10,22)(11,23)(12,20)(14,33,36,57)(15,18,28,42)(16,35,40,58)(24,47)(25,44)(26,45)(27,46)(29,32)(30,49)(31,48)(37,41,51,62)(38,56,59,67)(39,43,54,63)(50,65)(52,55)(53,64)(60,61,66,68);(5,8,10,26,11,27,30,53,12,24,31,50,9,25,29,52)(6,14,16,38,15,37,39,60,17,36,40,59,28,51,54,66)(7,19,22,45,23,46,49,64,20,47,48,65,21,44,32,55)(13,33,35,56,18,41,43,61,34,57,58,67,42,62,63,68);(7,20)(8,24)(13,34)(14,36)(18,42)(21,23)(22,48)(25,27)(26,50)(32,49)(35,58)(37,51)(38,59)(43,63)(52,53)(60,66)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=h,h^2=i,b^a=be,e^a=ei,g^a=gi,d^b=di,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, 400171);