#323149

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	#10144	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

721 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)(3,10)(4,6)(5,9)(7,11)(8,13)(12,14)(15,16)(17,18,22,27,23,20,30,26)(19,25,28,24,29,32,21,31);(2,6,9,8)(7,14,15,10)(17,19,22,28,23,29,30,21)(18,24,26,25,20,31,27,32);(1,2)(3,7)(4,8)(5,9)(6,13)(10,11)(12,15)(14,16)(17,20)(18,23)(19,24)(21,32)(22,27)(25,28)(26,30)(29,31);(1,3,4,11,5,12,13,16)(2,7,8,10,9,15,6,14)(17,21,22,19,23,28,30,29)(18,25,26,31,20,32,27,24)

Polycyclic presentation:

2^9|a^2=f,b^2=egh,d^2=e,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, 230345);