#10195685

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

4901 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10102881);