#121033

order	512
nilpotency class	3
isoclinism	isoclinic class of #121030
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	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

573 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,e^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,f^b=fi,g^b=gh,e^c=ei,g^c=ghi

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, 28229);