#382890

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 290086);