#383123

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
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×#3013	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1900 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290319);