#7660690

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	#37649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4828 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=h,e^2=g,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, 7567886);