#238061

order	512
nilpotency class	3
isoclinism	isoclinic class of #238058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
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

918 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 178 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)(19,24)(20,32,34,43)(21,27)(22,37)(23,26)(25,35,42,36)(28,41,44,48)(29,40)(30,39)(31,46)(33,38,45,47);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19)(18,24)(20,33,34,45)(21,30)(22,31)(23,29)(25,41,42,48)(26,40)(27,39)(28,36,44,35)(32,47,43,38)(37,46);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,20)(18,25)(19,28)(21,36)(22,35)(23,34)(24,38)(26,42)(27,43)(29,44)(30,33)(31,45)(32,37)(39,48)(40,47)(41,46);(18,26)(19,29)(25,42)(27,37)(28,44)(30,31)(32,43)(33,45)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,e^c=eh,f^c=fh,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, 145257);