#416770

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

932 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:

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

Polycyclic presentation:

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

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, 323966);