#369189

order	512
nilpotency class	3
isoclinism	isoclinic class of #220200
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3154	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1860 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=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, 276385);