#272922

order	512
nilpotency class	3
isoclinism	isoclinic class of #272795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#1598	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1227 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180118);