#232912

order	512
nilpotency class	3
isoclinism	isoclinic class of #232905
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1144 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1144 conjugacy classes of subgroups — too many to draw. Only its 198 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)(3,11)(4,6)(5,9)(7,10)(8,12)(13,16)(14,15)(17,18,23,28)(19,29,32,39)(20,33,36,42)(21,26,22,27)(24,37,40,31)(25,34,43,35)(30,38,46,41)(44,47,48,45);(1,3,6,13)(2,8,4,15)(5,14,7,12)(9,11,10,16)(17,19,23,32)(18,24,28,40)(20,30,36,46)(21,37,22,31)(25,41,43,38)(26,29,27,39)(33,44,42,48)(34,47,35,45);(1,4,6,2)(3,11,13,16)(5,9,7,10)(8,12,15,14)(17,20,23,36)(18,25,28,43)(19,30,32,46)(21,35,22,34)(24,38,40,41)(26,42,27,33)(29,44,39,48)(31,47,37,45);(1,5,6,7)(2,9,4,10)(3,12,13,14)(8,16,15,11)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,34,36,35)(24,39,40,29)(25,33,43,42)(30,45,46,47)(38,44,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140108);