#273154

order	512
nilpotency class	3
isoclinism	isoclinic class of #273051
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	#37693	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1444 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)(7,12,9,11)(8,10)(13,14)(15,20)(16,22)(17,19)(18,21)(23,28)(24,26)(25,27);(1,2)(3,4)(5,7)(6,11)(8,12)(9,10)(14,19)(16,17)(20,26)(21,22)(23,24)(27,28);(2,4)(5,8,10,6)(7,12,9,11)(13,15,18,25)(14,20,21,27)(16,23,17,24)(19,26,22,28);(5,9,10,7)(6,12,8,11)(13,16)(14,21)(15,23)(17,18)(20,27)(24,25)

Polycyclic presentation:

2^9|a^2=fhi,c^2=h,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dei,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, 180350);