#422535

order	512
nilpotency class	3
isoclinism	isoclinic class of #421157
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15307
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1206 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)(3,13,18,24)(4,19,23,27)(5,6)(7,11)(8,12)(9,16,15,17)(10,21,28,22)(14,25)(20,26)(29,31)(30,32);(1,3,7,16,8,18,6,17)(2,9,11,13,12,15,5,24)(4,20,21,14,23,30,22,31)(10,26,19,25,28,32,27,29);(1,4,6,21,8,23,7,22)(2,10,5,19,12,28,11,27)(3,14,16,20,18,31,17,30)(9,25,13,26,15,29,24,32);(1,5,6,12,8,11,7,2)(3,15,16,24,18,9,17,13)(4,19,21,28,23,27,22,10)(14,29,20,32,31,25,30,26)

Polycyclic presentation:

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

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, 329731);