#331303

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
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

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

normal subgroups only

This group has 698 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)(9,10,12,17,13,16,20,15)(11,14,19,21,18,22,24,23)(25,26)(27,34)(28,38)(29,32)(30,33)(31,36)(35,39)(37,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,21)(20,24)(25,27,28,35,29,36,30,37)(26,31,38,40,32,34,33,39);(1,3)(2,6)(4,7)(5,8)(10,15)(11,18)(12,20)(14,21)(16,17)(22,23)(26,32)(27,35)(28,30)(31,39)(34,40)(36,37);(3,7)(6,8)(10,16)(11,18)(15,17)(19,24)(25,28,29,30)(26,33,32,38)(27,35,36,37)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=ghi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,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, 238499);