#494920

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

778 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)(5,6,11,18,13,19,12,35)(7,20,24,39,26,48,25,55)(8,27,31,58,28,57,32,43)(9,17,33,37,34,45,10,16)(14,36,40,54,41,23,47,22)(15,29,44,60,42,59,61,30)(21,38,52,64,49,63,53,66)(46,50,56,62,65,67,68,51);(5,7,9,22,11,24,33,36,13,26,34,54,12,25,10,23)(6,14,37,48,35,47,17,39,19,41,16,20,18,40,45,55)(8,21,29,50,31,52,60,62,28,49,59,67,32,53,30,51)(15,38,58,65,61,66,27,56,42,63,43,46,44,64,57,68);(2,4)(5,8,9,29,11,31,33,60,13,28,34,59,12,32,10,30)(6,15,16,43,35,61,45,57,19,42,37,58,18,44,17,27)(7,21,22,50,24,52,36,62,26,49,54,67,25,53,23,51)(14,38,20,46,47,66,55,68,41,63,48,65,40,64,39,56);(8,28)(15,42)(21,49)(27,57)(29,59)(30,60)(31,32)(38,63)(43,58)(44,61)(46,65)(50,67)(51,62)(52,53)(56,68)(64,66)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=e,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,h^a=hi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402116);