#6539853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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 ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2065 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6447049);