#331873

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

690 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,f^a=fh,c^b=cg,d^b=deh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 239069);