#6533514

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3347 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^c=ei,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, 6440710);