#6552638

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×#3319	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1949 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,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, 6459834);