#272960

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fh,c^2=h,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,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, 180156);