#234372

order	512
nilpotency class	3
isoclinism	isoclinic class of #234365
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1386 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 198 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)(3,6)(4,7)(5,8)(9,14)(10,13)(11,15)(12,16)(17,18,22,26)(19,27,30,42)(20,24,31,40)(21,33,36,25)(23,34,38,29)(28,44,45,48)(32,43,35,41)(37,46,39,47);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19,22,30)(18,23,26,38)(20,28,31,45)(21,34,36,29)(24,39,40,37)(25,42,33,27)(32,47,35,46)(41,48,43,44);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,20)(18,24)(19,28)(21,35)(22,31)(23,37)(25,43)(26,40)(27,44)(29,47)(30,45)(32,36)(33,41)(34,46)(38,39)(42,48);(20,31)(24,40)(28,45)(32,35)(37,39)(41,43)(44,48)(46,47)

Polycyclic presentation:

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

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, 141568);