#272741

order	512
nilpotency class	3
isoclinism	isoclinic class of #272667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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 ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1442 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)(7,11,12,9)(8,10)(13,14,18,22)(15,20,17,19)(16,24,25,27)(21,26,28,23);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(15,17)(20,22)(21,24)(23,26)(27,28);(2,4)(5,6)(7,9)(8,10)(11,12)(13,15,18,17)(14,20,22,19)(16,23,25,26)(21,27,28,24);(5,7)(6,9)(8,12)(10,11)(13,16,18,25)(14,21,22,28)(15,23,17,26)(19,24,20,27)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^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, 179937);