#484735

order	512
nilpotency class	4
isoclinism	isoclinic class of #484583
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×C₂²⋊C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 748 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) = 128

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,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, 391931);