#6591777

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37846	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6498973);