#490634

order	512
nilpotency class	4
isoclinism	isoclinic class of #490474
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

456 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(18,23)(20,24)(21,31)(25,33)(26,41)(27,36)(29,37)(30,43)(32,38)(34,35)(39,44)(40,48)(42,47)(45,46);(2,6)(7,15)(8,13)(9,16)(17,18,24,35,21,26,32,36,22,27,38,41,31,34,20,23)(19,25,37,46,30,40,42,44,28,39,47,48,43,45,29,33);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,19)(18,25)(20,29)(21,30)(22,28)(23,33)(24,37)(26,40)(27,39)(31,43)(32,42)(34,45)(35,46)(36,44)(38,47)(41,48);(19,28)(25,39)(29,42)(30,43)(33,44)(37,47)(40,45)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397830);