#466921

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

868 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=gi,c^2=f,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi

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, 374117);