#232893

order	512
nilpotency class	3
isoclinism	isoclinic class of #232889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1148 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1148 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,12)(4,14)(5,16)(6,10)(7,11)(8,9)(13,15)(17,18,23,26)(19,29,31,43)(20,33,36,42)(21,28,22,27)(24,37,39,32)(25,34,41,35)(30,38,45,40)(44,47,48,46);(1,3)(2,8)(4,10)(5,14)(6,13)(7,9)(11,15)(12,16)(17,19)(18,24)(20,30)(21,37)(22,32)(23,31)(25,40)(26,39)(27,43)(28,29)(33,44)(34,47)(35,46)(36,45)(38,41)(42,48);(1,4,6,8)(2,9,10,14)(3,11,13,16)(5,15,7,12)(17,20,23,36)(18,25,26,41)(19,30,31,45)(21,35,22,34)(24,38,39,40)(27,33,28,42)(29,44,43,48)(32,47,37,46);(2,10)(3,13)(9,14)(11,16)(18,26)(19,31)(25,41)(27,28)(30,45)(32,37)(33,42)(46,47)

Polycyclic presentation:

2^9|a^2=i,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi

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, 140089);