#238086

order	512
nilpotency class	3
isoclinism	isoclinic class of #238082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 928 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)(19,23)(20,30,33,24)(21,34)(22,26)(25,35)(27,40,31,36)(28,42)(29,38)(32,39,46,44)(37,43)(41,47,45,48);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19)(18,23)(20,31,33,27)(21,28)(22,29)(24,36,30,40)(25,37)(26,38)(32,45,46,41)(34,42)(35,43)(39,47,44,48);(1,4,5,11)(2,7,8,15)(3,9,10,16)(6,12,13,14)(17,20)(18,24)(19,27)(21,32)(22,33)(23,36)(25,39)(26,30)(28,41)(29,31)(34,44)(35,46)(37,47)(38,40)(42,48)(43,45);(17,21,22,35)(18,25,26,34)(19,28,29,43)(20,32,33,46)(23,37,38,42)(24,39,30,44)(27,41,31,45)(36,47,40,48)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,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, 145282);