#465630

order	512
nilpotency class	3
isoclinism	isoclinic class of #465630
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16526
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1079 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)(3,13)(4,18)(5,11)(6,9)(7,12)(8,16)(10,21)(14,23)(15,27)(17,24)(19,26)(20,25)(22,30)(28,31)(29,32);(1,3,7,17,5,16,6,14)(2,8,12,23,11,13,9,24)(4,15,20,29,21,30,19,28)(10,22,26,31,18,27,25,32);(2,9)(3,14)(6,7)(8,13)(10,25)(11,12)(15,28)(16,17)(18,26)(19,20)(22,27)(29,30);(1,4,6,19,5,21,7,20)(2,10,12,26,11,18,9,25)(3,15,14,28,16,30,17,29)(8,22,23,31,13,27,24,32)

Polycyclic presentation:

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

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, 372826);