#459410

order	512
nilpotency class	3
isoclinism	isoclinic class of #459410
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16340
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

1154 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366606);