#215138

order	512
nilpotency class	3
isoclinism	isoclinic class of #213537
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	C₂²×#127	C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1426 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1426 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 122334);