#6448842

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23426	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3312 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)(7,10)(8,9)(11,12)(13,14,20,23)(15,24,26,27)(16,21,28,25)(17,18,22,19);(2,4)(13,15)(14,16)(17,24)(18,25)(19,21)(20,26)(22,27)(23,28);(6,9)(10,12)(13,16)(14,21)(15,17)(18,24)(19,27)(20,28)(22,26)(23,25);(5,7,8,11)(6,10,9,12)(13,17)(14,18)(15,16)(19,23)(20,22)(21,24)(25,27)(26,28);(13,18,20,19)(14,22,23,17)(15,25,26,21)(16,27,28,24)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356038);