#383488

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×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×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1397 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)(5,6,10,17,11,15,9,16)(7,12,19,28,20,14,26,33)(8,21,24,31,25,13,23,32)(18,30,22,36,34,27,35,29);(1,2)(3,4)(5,7)(6,12)(8,22)(9,26)(10,19)(11,20)(13,29)(14,15)(16,33)(17,28)(18,23)(21,36)(24,34)(25,35)(27,31)(30,32);(5,8,11,25)(6,13,15,21)(7,18,20,34)(9,24,10,23)(12,27,14,30)(16,32,17,31)(19,35,26,22)(28,36,33,29);(1,3,4,2)(5,7)(6,14)(8,18)(9,19)(10,26)(11,20)(12,15)(13,30)(16,33)(17,28)(21,27)(22,23)(24,35)(25,34)(29,32)(31,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^c=hi,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, 290684);