#10445007

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4937 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10352203);