#7660548

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	#42445	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4676 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7567744);