#331150

order	512
nilpotency class	3
isoclinism	isoclinic class of #331114
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 162 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 162 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)(9,10,14,16,13,18,12,17)(11,19,21,15,20,24,22,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,40)(35,39)(36,38);(2,5)(6,8)(11,20)(15,23)(19,24)(21,22)(25,27,33,34,28,35,29,36)(26,30,37,40,31,39,32,38);(1,3)(2,6)(4,7)(5,8)(9,11,13,20)(10,15,18,23)(12,21,14,22)(16,19,17,24)(26,31)(27,34)(29,33)(30,38)(35,36)(39,40);(3,7)(6,8)(9,12,13,14)(10,16,18,17)(11,21,20,22)(15,19,23,24)

Polycyclic presentation:

2^9|a^2=fhi,b^2=ghi,c^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,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, 238346);