#232369

order	512
nilpotency class	3
isoclinism	isoclinic class of #232351
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#34641	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1728 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1728 conjugacy classes of subgroups — too many to draw. Only its 258 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)(6,10)(7,8)(11,12)(13,14)(15,25)(16,21)(17,22)(18,23)(19,24)(20,27)(26,41)(28,36)(29,35)(30,34)(31,38)(32,40)(33,39)(37,42)(43,44);(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,20,24,36)(16,30,32,43)(17,28,18,27)(21,37,40,41)(22,25,23,35)(26,33,42,31)(34,39,44,38);(2,4)(6,10)(7,8)(11,12)(13,16)(14,21)(15,26)(17,31)(18,33)(19,32)(20,34)(22,38)(23,39)(24,40)(25,41)(27,30)(28,43)(29,42)(35,37)(36,44);(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,32,33)(20,35,36,25)(21,38,40,39)(26,30,42,43)(34,37,44,41)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139565);