#244466

order	512
nilpotency class	3
isoclinism	isoclinic class of #238015
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2398
prime radical	C₂³	#525
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

411 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=i,d^2=efi,f^2=h,g^2=hi,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=di,g^b=gi,e^c=eh,f^c=fh,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151662);