#331319

order	512
nilpotency class	3
isoclinism	isoclinic class of #331304
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	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

746 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 746 conjugacy classes of subgroups — too many to draw. Only its 154 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)(9,10,12,16,13,18,14,17)(11,19,21,23,20,15,22,24)(25,26,28,33,29,32,34,31)(27,30,37,39,36,38,35,40);(2,5)(6,8)(9,11)(10,15)(12,21)(13,20)(14,22)(16,24)(17,23)(18,19)(25,27,34,35,29,36,28,37)(26,30,31,40,32,38,33,39);(1,3)(2,6)(4,7)(5,8)(10,16)(11,20)(12,14)(15,23)(17,18)(19,24)(26,31)(27,35)(28,34)(30,38)(32,33)(36,37);(3,7)(6,8)(9,12,13,14)(10,17,18,16)(11,21,20,22)(15,23,19,24)(26,32)(27,36)(31,33)(35,37)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238515);