#403117

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1200 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1200 conjugacy classes of subgroups — too many to draw. Only its 210 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efi,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,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, 310313);