#405507

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1492 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=e,b^2=ehi,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 312703);