#6607185

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2323 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:

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

Polycyclic presentation:

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

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, 6514381);