#352245

order	512
nilpotency class	3
isoclinism	isoclinic class of #352238
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19267	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(18,19)(20,28)(21,22)(24,31)(25,26)(29,30);(2,6)(7,14)(8,13)(9,15)(17,18,22,25,23,26,21,19)(20,24,29,32,30,31,28,27);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,19,21,26,23,25,22,18)(20,27,28,31,30,32,29,24);(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,32)(26,31)

Polycyclic presentation:

2^9|b^2=ei,c^2=fh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,g^d=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259441);