#318600

order	512
nilpotency class	3
isoclinism	isoclinic class of #318583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#547
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

511 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

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, 225796);