#352316

order	512
nilpotency class	3
isoclinism	isoclinic class of #352310
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

660 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)(17,18,24,19)(20,22,26,23)(21,28,31,29)(25,32,27,30);(2,6)(7,14)(8,13)(9,15)(17,19,22,26,24,18,23,20)(21,27,30,29,31,25,32,28);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,20,22,19,24,26,23,18)(21,29,30,25,31,28,32,27);(3,10)(7,14)(9,15)(11,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,32)(24,31)(26,29)

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^c=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, 259512);