#381134

order	512
nilpotency class	3
isoclinism	isoclinic class of #214664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1080 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1080 conjugacy classes of subgroups — too many to draw. Only its 242 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=g,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,h^b=hi,d^c=de,h^c=hi,f^d=fi

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, 288330);