#112271

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1104 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1104 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)(3,14,16,28)(4,18,17,9)(5,20)(6,12)(7,13)(8,15,25,32)(10,22)(11,24)(19,27,21,26)(23,29)(30,31);(1,3,6,4)(2,8,12,26)(5,21,22,32)(7,17,24,16)(9,20,28,10)(11,25,13,27)(14,29,18,30)(15,31,19,23);(1,4,6,3)(2,9,12,28)(5,15,22,19)(7,16,24,17)(8,10,26,20)(11,18,13,14)(21,31,32,23)(25,29,27,30)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^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, 19467);