#348304

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
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

829 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 829 conjugacy classes of subgroups — too many to draw. Only its 164 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)(5,6,10,14)(7,15,11,12)(8,18,19,17)(9,20,13,16)(21,22,26,25)(23,30,27,28)(24,32,34,36)(29,35,33,31);(1,2,3,4)(5,7,10,11)(6,12,14,15)(8,16,19,20)(9,18,13,17)(21,23,26,27)(22,28,25,30)(24,31,34,35)(29,32,33,36);(5,8)(6,13)(7,16)(9,14)(10,19)(11,20)(12,17)(15,18)(21,24,26,34)(22,29,25,33)(23,31,27,35)(28,32,30,36);(5,9,10,13)(6,8,14,19)(7,17,11,18)(12,20,15,16)(21,25,26,22)(23,28,27,30)(24,33,34,29)(31,32,35,36)

Polycyclic presentation:

2^9|a^2=h,b^2=e,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,d^b=dh,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, 255500);