#112277

order	512
nilpotency class	3
isoclinism	isoclinic class of #112268
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1114 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1114 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=i,g^2=hi,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, 19473);