#486675

order	512
nilpotency class	4
isoclinism	isoclinic class of #486312
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

692 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,6,10,13)(4,7,11,14)(9,15,16,12)(18,25)(19,30)(20,34)(21,36)(22,23)(26,41)(27,29)(28,35)(31,32)(33,38)(37,39)(40,43)(42,44)(45,48);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,18,24,29)(19,26,33,45)(20,35,21,37)(22,28,23,39)(25,34,27,36)(30,43,38,44)(31,41,32,48)(40,46,42,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,16)(13,15)(17,19)(18,26)(20,31)(21,32)(22,38)(23,30)(24,33)(25,40)(27,42)(28,44)(29,45)(34,46)(35,41)(36,47)(37,48)(39,43);(17,20,22,36,24,21,23,34)(18,27,39,37,29,25,28,35)(19,31,38,47,33,32,30,46)(26,42,43,48,45,40,44,41)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393871);