#326512

order	512
nilpotency class	3
isoclinism	isoclinic class of #326502
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2563
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

414 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eghi,c^2=i,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^b=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233708);