#361405

order	512
nilpotency class	3
isoclinism	isoclinic class of #188774
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

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	#2283
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

721 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)(7,17,20,29)(8,13,22,12)(9,23,25,36)(10,15)(11,16)(14,24,30,32)(18,26)(19,28)(21,34)(27,33)(31,35);(1,2)(3,4)(5,7,18,22,11,20,10,8)(6,12,15,17,16,13,26,29)(9,19,32,35,25,33,24,21)(14,27,23,31,30,28,36,34);(2,4)(5,8,10,20,11,22,18,7)(6,13,15,29,16,12,26,17)(9,21,24,33,25,35,32,19)(14,28,23,34,30,27,36,31);(5,9,11,25)(6,14)(7,19,20,33)(8,21,22,35)(10,24,18,32)(12,27)(13,28)(15,23)(16,30)(17,31)(26,36)(29,34)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,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, 268601);