#433835

order	512
nilpotency class	3
isoclinism	isoclinic class of #433835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15493
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1223 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,g^d=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341031);