#492948

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂²	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1302 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400144);