#498114

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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	#3002	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₂²⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,d^b=dei,e^b=ei,e^d=ei

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, 405310);