#433831

order	512
nilpotency class	3
isoclinism	isoclinic class of #433831
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	#23402	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1260 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,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, 341027);