#484844

order	512
nilpotency class	4
isoclinism	isoclinic class of #484587
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	#2619	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 813 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)(19,27)(20,25)(21,23)(22,26)(24,30)(28,41)(29,43)(31,42)(32,39)(33,37)(34,36)(35,38)(40,45)(44,48)(46,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,23)(20,31)(24,36)(25,40)(26,38)(27,43)(28,32)(33,45)(37,42)(39,48)(41,47)(44,46);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,19)(18,24,26,27)(20,32)(21,34)(22,30)(23,36,38,43)(25,41,42,39)(28,31)(29,35)(33,44)(37,48,40,47)(45,46);(17,20,22,31)(18,25,26,42)(19,28,30,32)(21,33,35,45)(23,37,38,40)(24,39,27,41)(29,44,34,46)(36,47,43,48)

Polycyclic presentation:

2^9|a^2=g,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,d^c=di,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, 392040);