#242959

order	512
nilpotency class	3
isoclinism	isoclinic class of #10631
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

612 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 202 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)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18,22,27,23,28,21,26)(19,29,31,24,33,40,32,41)(20,34,36,42,37,43,38,25)(30,45,47,48,35,39,46,44);(1,3,4,10)(2,6,7,14)(5,9,12,11)(8,16,15,13)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,39)(26,40)(27,29)(28,41)(30,37)(34,48)(36,47)(38,46)(42,45)(43,44);(2,7)(3,9)(5,12)(6,13)(10,11)(14,16)(17,20,21,36,23,37,22,38)(18,25,26,34,28,42,27,43)(19,30,32,47,33,35,31,46)(24,39,29,44,41,45,40,48);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,40,41,29)(25,34,42,43)(30,46,35,47)(39,48,45,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150155);