#498385

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,6)(7,8)(9,10,14,18,13,19,28,36,15,20,30,37,29,17,33,35)(11,12,22,25,21,26,32,39,23,27,34,40,38,24,16,31);(1,2)(3,7)(4,8)(5,6)(9,11,14,22,13,21,28,32,15,23,30,34,29,38,33,16)(10,12,18,25,19,26,36,39,20,27,37,40,17,24,35,31);(1,3,5,4)(2,7,6,8)(9,12)(10,16)(11,18)(13,26)(14,25)(15,27)(17,34)(19,22)(20,32)(21,36)(23,37)(24,29)(28,39)(30,40)(31,33)(35,38);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,f^c=fi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405581);