#498115

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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

747 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)(8,12,23,16,24,29,32,34,21,28,35,33,22,31,36,20);(5,7,10,14,9,18,25,15,11,19,27,13,26,17,30,6)(8,20,23,12,24,16,32,29,21,34,35,28,22,33,36,31);(2,4)(8,21)(12,28)(16,33)(20,34)(22,24)(23,35)(29,31)(32,36);(5,8)(6,12)(7,16)(9,24)(10,23)(11,21)(13,31)(14,29)(15,28)(17,20)(18,34)(19,33)(22,26)(25,32)(27,35)(30,36)

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,d^c=di,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405311);