#416452

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

960 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:

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

Polycyclic presentation:

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

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, 323648);