#383216

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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×#3012	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1969 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:

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

Polycyclic presentation:

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

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, 290412);