#484339

order	512
nilpotency class	4
isoclinism	isoclinic class of #484176
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

780 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=di,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi

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, 391535);