#249366

order	512
nilpotency class	3
isoclinism	isoclinic class of #249355
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1164 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1164 conjugacy classes of subgroups — too many to draw. Only its 152 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)(3,7,5,9)(4,8,13,16)(11,15,12,14)(17,18)(19,23)(20,30)(21,33)(22,26)(24,32)(25,36)(27,44)(28,42)(29,39)(31,43)(34,38)(35,41)(37,46)(40,47)(45,48);(1,3,6,5)(2,7,10,9)(4,11,13,12)(8,14,16,15)(17,19,22,29)(18,23,26,39)(20,27,31,45)(21,34,36,28)(24,40,41,37)(25,42,33,38)(30,44,43,48)(32,47,35,46);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,20)(18,24)(19,27)(21,35)(22,31)(23,37)(25,43)(26,41)(28,47)(29,45)(30,33)(32,36)(34,46)(38,48)(39,40)(42,44);(1,5,6,3)(2,9,10,7)(4,12,13,11)(8,15,16,14)(20,31)(24,41)(27,45)(30,43)(32,35)(37,40)(44,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156562);