#352327

order	512
nilpotency class	3
isoclinism	isoclinic class of #352320
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	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

644 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=f,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,c^b=ch,d^b=di,g^d=gh

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, 259523);