#272707

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
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	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1329 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179903);