#232372

order	512
nilpotency class	3
isoclinism	isoclinic class of #232351
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1332 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1332 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)(5,6,10,7)(8,11,9,12)(13,14,19,24)(15,25,29,35)(16,21,32,40)(17,22,18,23)(20,28,36,27)(26,41,42,37)(30,34,43,44)(31,38,33,39);(6,11)(7,12)(8,9)(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)(5,7,10,6)(8,12,9,11)(13,16,19,32)(14,21,24,40)(15,26,29,42)(17,31,18,33)(20,34,36,44)(22,38,23,39)(25,41,35,37)(27,30,28,43);(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|a^2=h,b^2=hi,c^2=h,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, 139568);