#6579314

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	#37680	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2035 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)(17,21,24,25)(18,22,19,23);(1,2)(3,4)(6,9)(10,12)(13,15,19,23,20,14,18,22)(16,17,21,28,27,24,25,26);(2,4)(5,7,8,11)(6,10,9,12)(13,16,19,21,20,27,18,25)(14,17,22,28,15,24,23,26);(1,2)(3,4)(13,17)(14,21)(15,25)(16,23)(18,26)(19,28)(20,24)(22,27);(2,4)(13,18,20,19)(14,22,15,23)(16,25,27,21)(17,28,24,26)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,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,d^b=di,e^b=eh,d^c=dh,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486510);