#331216

order	512
nilpotency class	3
isoclinism	isoclinic class of #331190
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

694 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 178 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)(3,6)(4,5)(7,8)(9,10,14,18,13,17,12,16)(11,19,21,24,20,15,22,23)(25,26,29,33,30,34,35,39)(27,31,37,32,28,38,36,40);(2,5)(6,8)(9,11)(10,15)(12,22)(13,20)(14,21)(16,24)(17,19)(18,23)(25,27,35,36,30,28,29,37)(26,31,39,40,34,38,33,32);(1,3)(2,6)(4,7)(5,8)(10,16)(11,20)(12,14)(15,23)(17,18)(19,24)(25,28,30,27)(26,32,34,40)(29,37,35,36)(31,39,38,33);(3,7)(6,8)(9,12,13,14)(10,16,17,18)(11,21,20,22)(15,23,19,24)(25,29,30,35)(26,33,34,39)(27,36,28,37)(31,40,38,32)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gi

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, 238412);