#6607081

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2374 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)(13,14,19,23)(15,24,25,16)(17,22,18,28)(20,21,27,26);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,21,19,25,18,26)(14,20,28,16,23,27,22,24);(7,12)(9,10)(13,16,18,20,19,24,17,27)(14,21,22,15,23,26,28,25);(5,7)(6,10)(8,12)(9,11)(15,25)(16,24)(20,27)(21,26);(7,12)(9,10)(13,17,19,18)(14,22,23,28)(15,26,25,21)(16,27,24,20)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,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,d^b=di,e^b=ehi,d^c=dhi,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6514277);