#331294

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29806	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 765 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)(29,33)(30,32)(31,36)(35,40)(37,39);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,21)(20,24)(25,27,30,37,29,36,28,35)(26,31,32,40,33,34,38,39);(1,3)(2,6)(4,7)(5,8)(10,15)(11,18)(12,20)(14,21)(16,17)(22,23)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(3,7)(6,8)(10,16)(11,18)(15,17)(19,24)(25,28,29,30)(26,32,33,38)(27,35,36,37)(31,40,34,39)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,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, 238490);