#361381

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1118 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)(7,15,18,28)(8,19,21,12)(9,13)(10,14)(11,17,24,32)(16,29)(20,34,35,25)(22,26)(23,30)(27,31)(33,36);(1,2)(3,4)(5,7)(6,11,14,24)(8,16)(9,17)(10,18)(12,23,19,33)(13,15,26,28)(20,30)(21,31)(22,32)(25,27,34,29)(35,36);(2,4)(5,8,9,20,10,21,22,35)(6,12,13,25,14,19,26,34)(7,16,17,30,18,31,32,36)(11,23,15,27,24,33,28,29);(5,7)(6,11,14,24)(8,16)(9,17)(10,18)(12,23,19,33)(13,15,26,28)(20,30)(21,31)(22,32)(25,27,34,29)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268577);