#491449

order	512
nilpotency class	4
isoclinism	isoclinic class of #12503
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

570 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 174 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)(20,40)(21,28)(22,30)(23,29)(24,31)(25,32)(26,36)(27,39)(34,47)(35,48)(37,44)(38,45)(41,43)(42,46);(1,3,4,10,5,11,13,16)(2,6,8,9,7,14,12,15)(17,19,34,42,21,35,22,36,25,39,23,37,24,38,41,20)(18,26,43,48,31,46,29,33,32,40,30,45,28,44,47,27);(2,7)(6,14)(8,12)(9,15)(17,20,41,38,24,37,23,39,25,36,22,35,21,42,34,19)(18,27,47,44,28,45,30,40,32,33,29,46,31,48,43,26);(17,21,25,24)(18,28,32,31)(19,35,39,38)(20,42,36,37)(22,23,41,34)(26,44,40,46)(27,45,33,48)(29,43,47,30)

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225180, #72225200
maximal normal subgroup of	#72225180, #72225200

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398645);