#6514539

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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	#35521	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=f,d^2=hi,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=dhi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6421735);