#362566

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2285 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=e,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 269762);