#249384

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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1219 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1219 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) = 64

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=hi,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, 156580);