#359392

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=efh,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^c=ei,g^c=gi,h^c=hi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266588);