#6559035

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535648
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2381 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^d=ehi

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, 6466231);