#485843

order	512
nilpotency class	4
isoclinism	isoclinic class of #485617
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	#392	C₂³
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

756 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 120 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)(17,18,22,26)(19,27,28,42)(20,24,33,41)(21,34,36,25)(23,35,37,30)(29,43,31,45)(32,47,48,40)(38,46,39,44);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,23)(20,31)(21,35)(22,28)(24,39)(25,42)(26,37)(27,34)(29,33)(30,36)(32,44)(38,41)(40,43)(45,47)(46,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,28)(21,36)(23,37)(27,42)(29,31)(32,48)(38,39)(43,45);(3,12)(7,16)(10,15)(11,14)(17,20,22,33)(18,24,26,41)(19,29,28,31)(21,32,36,48)(23,38,37,39)(25,40,34,47)(27,43,42,45)(30,44,35,46)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393039);