#6461371

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,28)(22,27)(23,25)(24,29)(26,31)(30,32);(2,5)(6,7)(9,11,12,15)(10,13,14,16)(17,19,21,18,23,27,22,25)(20,26,30,24,29,32,31,28);(1,3,4,8)(2,6,5,7)(10,14)(11,15)(17,20,23,29)(18,24,25,28)(19,26,27,32)(21,30,22,31);(11,15)(13,16)(20,29)(24,28)(26,32)(30,31);(17,21,23,22)(18,19,25,27)(20,30,29,31)(24,26,28,32)

Polycyclic presentation:

2^9|b^2=fg,c^2=gh,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di

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, 6368567);