#359598

order	512
nilpotency class	3
isoclinism	isoclinic class of #184991
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3489	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3856 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)(7,17,21,12)(8,22,25,32)(9,26,20,14)(10,16)(11,15)(13,24,31,27)(18,34)(19,29)(23,28)(30,35)(33,36);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,28)(14,29)(15,17)(16,26)(22,30)(23,24)(25,35)(27,36)(31,33)(32,34);(2,4)(5,8)(6,13)(7,18)(9,23)(10,27)(11,25)(12,28)(14,30)(15,31)(16,32)(17,33)(19,24)(20,36)(21,35)(22,29)(26,34);(5,9,11,20)(6,14,15,26)(7,19,21,10)(8,23,25,36)(12,29,17,16)(13,30,31,34)(18,24,35,27)(22,33,32,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266794);