#232432

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	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1358 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 1358 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14)(15,24)(16,21)(17,33)(18,22)(19,23)(20,26)(25,41)(27,35)(28,40)(29,34)(30,38)(31,37)(32,39)(36,43)(42,44);(6,10)(7,8)(11,12)(13,15,19,28)(14,20,23,35)(16,29,32,42)(17,27,18,26)(21,36,39,41)(22,40,33,24)(25,30,43,31)(34,38,44,37);(2,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,21)(15,25)(17,31)(18,30)(19,32)(20,34)(22,38)(23,39)(24,41)(26,29)(27,42)(28,43)(33,37)(35,44)(36,40);(6,11)(10,12)(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,30,32,31)(20,24,35,40)(21,37,39,38)(25,42,43,29)(34,36,44,41)

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,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, 139628);