#6577447

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	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2055 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)(15,16)(19,20)(23,26)(24,25);(5,7,9,8)(6,10,12,11)(13,14,17,21)(15,23,16,26)(18,28,27,22)(19,24,20,25);(2,4)(5,7,9,8)(6,11,12,10)(13,15,17,16)(14,19,21,20)(18,24,27,25)(22,26,28,23);(5,8,9,7)(6,10,12,11)(15,24)(16,25)(18,27)(19,26)(20,23)(22,28);(13,16,17,15)(14,20,21,19)(18,25,27,24)(22,23,28,26)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,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=ehi,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, 6484643);