#327934

order	512
nilpotency class	3
isoclinism	isoclinic class of #326835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

721 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 721 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 235130);