#6508822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500385
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
prime radical	#35000	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2199 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)(5,6,10,11)(7,8,12,9)(13,14,17,20)(15,18,22,26)(16,19,24,28)(21,25,23,27);(5,7,9,6,10,12,8,11)(13,15)(14,18)(16,23)(17,22)(19,27)(20,26)(21,24)(25,28);(2,4)(6,11)(7,12)(13,16)(14,19)(15,21)(17,24)(18,25)(20,28)(22,23)(26,27);(16,24)(19,28)(21,23)(25,27);(5,8,10,9)(6,12,11,7)(15,22)(18,26)(21,23)(25,27)

Polycyclic presentation:

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

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, 6416018);