#484833

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	C₂⁵	C₂×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

811 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 811 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ghi,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, 392029);