#323168

order	512
nilpotency class	3
isoclinism	isoclinic class of #323133
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19671	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230364);