#9674925

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

4690 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9582121);