#6577573

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#23929	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2078 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484769);