#249386

order	512
nilpotency class	3
isoclinism	isoclinic class of #249355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1244 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1244 conjugacy classes of subgroups — too many to draw. Only its 152 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156582);