#6533402

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3382 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)(13,14,19,17)(15,22,26,28)(16,21,18,27)(20,23,24,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,24)(18,23)(19,26)(21,28)(22,27);(2,4)(6,9)(10,12)(13,16,19,18)(14,21,17,27)(15,23,26,25)(20,22,24,28);(5,7)(6,10)(8,11)(9,12)(13,17,19,14)(15,24,26,20)(16,21,18,27)(22,25,28,23);(7,11)(10,12)(13,16,19,18)(14,21,17,27)(15,25,26,23)(20,28,24,22)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6440598);