#7733804

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
prime radical	#37185	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5016 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=g,e^2=h,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, 7641000);