#272995

order	512
nilpotency class	3
isoclinism	isoclinic class of #272923
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
prime radical	#23918	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1463 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)(13,14,17,21)(15,19,23,27)(16,18,24,20)(22,25,28,26);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,18)(16,24)(19,25)(20,21)(22,28)(26,27);(2,4)(5,6)(7,9)(8,10)(11,12)(13,15,17,23)(14,19,21,27)(16,22,24,28)(18,25,20,26);(5,7)(6,9)(8,12)(10,11)(14,20)(16,24)(18,21)(19,26)(22,28)(25,27)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,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, 180191);