#6584461

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#36434	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2143 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)(15,21)(16,22)(17,23)(18,20)(19,24)(25,27)(26,28);(5,7,9,8)(6,10,11,12)(14,20)(15,17)(16,19)(21,28)(23,27)(25,26);(1,2,3,4)(13,15)(14,21)(16,17)(18,25)(19,26)(20,27)(22,23)(24,28);(5,8,9,7)(6,10,11,12)(13,16)(14,22)(17,26)(18,19)(20,24)(23,28);(13,17,18,26)(14,23,20,28)(15,16,25,19)(21,22,27,24)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,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, 6491657);