#238547

order	512
nilpotency class	3
isoclinism	isoclinic class of #232415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1160 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1160 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gh,d^c=dhi,f^c=fh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145743);