#112270

order	512
nilpotency class	3
isoclinism	isoclinic class of #112268
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1206 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1206 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,g^b=ghi,d^c=dh,g^c=ghi

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, 19466);