#114277

order	512
nilpotency class	3
isoclinism	isoclinic class of #114273
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#60801	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

922 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 922 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21473);