#6583971

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#35831	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2114 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:

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

Polycyclic presentation:

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

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, 6491167);