#381133

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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1128 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^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, 288329);