#6682789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2340 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6589985);