#268719

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	#19496	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^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, 175915);