#6468153

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 6375349);