#10155625

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

5056 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=gi,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, 10062821);