#491453

order	512
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#493
prime radical	#19603	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

562 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398649);