#6629759

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
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	C₂×#3061	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2296 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,27)(17,22)(18,25)(19,23)(20,26)(21,28);(5,7,8,11)(6,9,10,12)(13,15,17,27,19,21,18,26)(14,20,25,28,23,16,22,24);(6,10)(9,12)(13,14)(15,20)(16,21)(17,25)(18,22)(19,23)(24,26)(27,28);(2,4)(13,16,18,28,19,20,17,24)(14,21,22,27,23,15,25,26);(13,17,19,18)(14,22,23,25)(15,26,21,27)(16,24,20,28)

Polycyclic presentation:

2^9|b^2=fgi,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,d^b=di,e^b=ei,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536955);