#7733558

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	#44892	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4906 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)(19,23)(20,29)(21,25)(22,31)(24,27)(26,30)(28,32);(1,2)(3,7)(4,5)(6,8)(9,10,13,14)(11,15,12,16)(17,19,24,28)(18,23,27,32)(20,30,21,22)(25,31,29,26);(2,5)(6,7)(17,20,24,21)(18,25,27,29)(19,22,28,30)(23,26,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,21,24,20)(18,25,27,29)(19,22,28,30)(23,31,32,26);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,22)(18,26)(19,20)(21,28)(23,25)(24,30)(27,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,16,14,15)(17,23)(18,19)(20,26)(21,31)(22,25)(24,32)(27,28)(29,30)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gh,e^2=g,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, 7640754);