#7737480

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

4913 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7644676);