#120812

order	512
nilpotency class	3
isoclinism	isoclinic class of #120812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

428 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 28008);