#6579320

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37683	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2047 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,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, 6486516);