#352016

order	512
nilpotency class	3
isoclinism	isoclinic class of #352010
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

899 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 899 conjugacy classes of subgroups — too many to draw. Only its 222 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:

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

Polycyclic presentation:

2^9|b^2=ehi,c^2=fh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259212);