#6539819

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#3307	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2033 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6447015);