#120988

order	512
nilpotency class	3
isoclinism	isoclinic class of #120987
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	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

417 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

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, 28184);