#327968

order	512
nilpotency class	3
isoclinism	isoclinic class of #326925
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

649 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 649 conjugacy classes of subgroups — too many to draw. Only its 132 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)(17,18,21,26)(19,20,29,24)(22,28,23,27)(25,31,32,30)(33,34)(35,43)(36,39)(37,41)(38,47)(40,45)(42,46)(44,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,29)(18,24,26,20)(22,25,23,32)(27,31,28,30)(33,35,37,36)(34,39,41,43)(38,45,46,44)(40,47,48,42);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20,21,24)(18,25,26,32)(19,28,29,27)(22,31,23,30)(33,36,37,35)(34,40,41,48)(38,44,46,45)(39,42,43,47);(4,12)(7,14)(10,16)(13,15)(19,29)(20,24)(25,32)(30,31)(35,36)(39,43)(40,48)(44,45)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,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, 235164);