#331925

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, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 612 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,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, 239121);