#512558

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

785 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)(5,6,8,13,9,14,21,31,10,15,22,32,24,34,43,53,11,16,23,33,25,35,44,54,26,36,45,55,46,56,61,66)(7,12,18,28,19,29,40,50,20,30,41,51,42,52,60,65,17,27,37,47,38,48,57,62,39,49,58,63,59,64,67,68);(2,4);(5,7)(6,12)(8,18)(9,19)(10,20)(11,17)(13,28)(14,29)(15,30)(16,27)(21,40)(22,41)(23,37)(24,42)(25,38)(26,39)(31,50)(32,51)(33,47)(34,52)(35,48)(36,49)(43,60)(44,57)(45,58)(46,59)(53,65)(54,62)(55,63)(56,64)(61,67)(66,68);(7,17)(12,27)(18,37)(19,38)(20,39)(28,47)(29,48)(30,49)(40,57)(41,58)(42,59)(50,62)(51,63)(52,64)(60,67)(65,68)

Polycyclic presentation:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419754);