#493348

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 400544);