#121119

order	512
nilpotency class	3
isoclinism	isoclinic class of #121117
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

612 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 28315);