#6502768

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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2275 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)(15,21)(16,25)(17,19)(18,24)(20,22)(23,27)(26,28);(5,6)(7,11)(8,10)(9,12)(13,15,22,28,17,23,16,24)(14,18,25,27,19,26,20,21);(2,4)(5,7)(6,9)(8,12)(10,11)(14,19)(15,23)(20,25)(24,28);(7,12)(9,11)(15,23)(18,26)(21,27)(24,28);(6,10)(9,11)(15,23)(18,26)(21,27)(24,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,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, 6409964);