#406149

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1016 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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,f^c=fi

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, 313345);