#215114

order	512
nilpotency class	3
isoclinism	isoclinic class of #195990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1068 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 226 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)(3,7)(4,6)(5,8)(9,10)(11,17,18,15)(12,20)(13,14)(16,21)(19,22,24,23)(25,26,29,34,30,32,28,33)(27,31,36,40,37,38,35,39);(1,3,4,8)(2,5,6,7)(10,14)(11,18)(16,20)(19,24)(25,27,30,37)(26,31,32,38)(28,35,29,36)(33,39,34,40);(3,8)(5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(26,32)(28,29)(31,38)(35,36);(2,6)(5,7)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,28,30,29)(26,33,32,34)(27,35,37,36)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^c=di,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122310);