#269515

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#59035	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1311 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 176711);