#213617

order	512
nilpotency class	3
isoclinism	isoclinic class of #163012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#121	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1360 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 218 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)(9,10,12,16,13,15,11,14)(17,18)(19,25,27,32)(20,24)(21,23)(22,26,30,28)(29,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16);(3,8)(5,7)(10,15)(11,12)(17,19)(18,22)(20,28)(21,27)(23,30)(24,32)(25,31)(26,29);(2,6)(5,7)(18,23)(22,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi

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, 120813);