#352910

order	512
nilpotency class	3
isoclinism	isoclinic class of #352019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

985 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=f,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260106);