#6578550

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	#24265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 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)(5,6)(7,11)(8,10)(9,12)(13,14,20,19)(15,21,17,22)(16,18,23,28)(24,27,26,25);(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,25)(17,20)(18,24)(19,22)(23,27)(26,28);(2,4)(13,16)(14,18)(15,24)(17,26)(19,28)(20,23)(21,27)(22,25);(5,7,9,8)(6,11,12,10)(13,17,20,15)(14,22,19,21)(16,25,23,27)(18,24,28,26);(5,8,9,7)(6,10,12,11)(13,18,20,28)(14,23,19,16)(15,25,17,27)(21,24,22,26)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,e^2=fi,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, 6485746);