#391712

order	512
nilpotency class	3
isoclinism	isoclinic class of #213744
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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	#35659	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1799 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1799 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

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

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, 298908);