#352943

order	512
nilpotency class	3
isoclinism	isoclinic class of #352047
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

953 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 953 conjugacy classes of subgroups — too many to draw. Only its 234 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,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^c=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, 260139);