#331028

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

Polycyclic presentation:

2^9|a^2=ei,c^2=i,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, 238224);