#268644

order	512
nilpotency class	3
isoclinism	isoclinic class of #268627
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61788	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

846 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,e^2=i,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=gi,e^d=ei,f^d=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175840);