#232478

order	512
nilpotency class	3
isoclinism	isoclinic class of #232469
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	C₂³×C₄	C₂⁴
prime radical	#60940	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1252 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 1252 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)(3,10)(4,7)(5,9)(6,13)(8,14)(11,15)(12,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,37)(30,46)(31,41)(33,40)(35,45)(36,44)(38,43)(39,48)(42,47);(1,3,5,12)(2,6,9,15)(4,11,14,13)(7,10,8,16)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,42)(26,29)(27,40)(28,41)(30,36)(34,39)(37,48)(38,47)(43,46)(44,45);(2,7)(3,11)(4,14)(8,9)(10,16)(12,13)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,39)(26,34)(27,43)(28,44)(29,45)(31,47)(32,35)(33,48)(40,42)(41,46);(3,12)(6,15)(10,16)(11,13)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,29,41,40)(25,34,44,43)(30,47,48,35)(39,45,46,42)

Polycyclic presentation:

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

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, 139674);