#331029

order	512
nilpotency class	3
isoclinism	isoclinic class of #331007
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

468 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 116 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)(3,9)(4,6)(5,8)(7,11)(10,15)(12,14)(13,16)(17,18,22,27,24,28,23,21)(19,29,31,40,32,43,38,25)(20,33,34,26,36,37,35,42)(30,39,45,48,46,41,47,44);(2,6)(3,10)(5,12)(7,15)(9,16)(11,13)(17,19)(18,25)(20,30)(21,29)(22,38)(23,31)(24,32)(26,41)(27,43)(28,40)(33,44)(34,47)(35,45)(36,46)(37,48)(39,42);(1,3,4,11)(2,7,6,9)(5,13,12,10)(8,15,14,16)(17,20,24,36)(18,26,28,42)(19,30,32,46)(21,37,27,33)(22,35,23,34)(25,39,40,41)(29,44,43,48)(31,47,38,45);(3,11)(7,9)(10,13)(15,16)(17,21,23,28,24,27,22,18)(19,29,31,40,32,43,38,25)(20,26,35,33,36,42,34,37)(30,41,45,44,46,39,47,48)

Polycyclic presentation:

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

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, 238225);