#6584501

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	#36051	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1926 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)(7,11)(8,12)(9,10)(13,14,18,19)(15,20,23,27)(16,21,25,28)(17,26,24,22);(6,10)(11,12)(14,19)(15,20)(17,22)(21,28)(23,27)(24,26);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,15,18,23)(14,20,19,27)(16,24,25,17)(21,22,28,26);(5,8,9,7)(6,12,10,11)(13,16,18,25)(14,21,19,28)(15,22,23,26)(17,27,24,20);(13,17,18,24)(14,22,19,26)(15,16,23,25)(20,28,27,21)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6491697);