#348274

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

811 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255470);