#269574

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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,d^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, 176770);