#327936

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

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

normal subgroups only

This group has 646 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)(19,24)(20,30)(21,26)(22,27)(23,28)(25,31)(29,32)(33,34,39,43)(35,40,36,45)(37,42,38,47)(41,44,48,46);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,24)(20,28)(21,29)(22,25)(23,30)(26,32)(27,31)(33,35)(34,40)(36,39)(37,44)(38,46)(41,47)(42,48)(43,45);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20,23,24)(18,25,28,29)(19,27,30,26)(21,31,22,32)(33,36,39,35)(34,41,43,48)(37,46,38,44)(40,47,45,42);(4,12)(7,14)(10,16)(13,15)(17,21,23,22)(18,26,28,27)(19,29,30,25)(20,31,24,32)(33,37,39,38)(34,42,43,47)(35,44,36,46)(40,48,45,41)

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=h,d^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, 235132);