#268692

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	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

781 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=h,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, 175888);