#232421

order	512
nilpotency class	3
isoclinism	isoclinic class of #232415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1584 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1584 conjugacy classes of subgroups — too many to draw. Only its 210 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,g^a=gi,c^b=cg,d^b=dhi,e^b=eh,g^b=gh,d^c=dh,e^c=ei,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, 139617);