#7660631

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	#37277	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4595 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)(17,18,24,23)(19,27,26,21)(20,30,22,25)(28,32,29,31);(1,2)(3,6)(4,7)(5,8)(9,10)(11,15)(12,16)(13,14)(17,19)(18,21)(20,28)(22,29)(23,27)(24,26)(25,32)(30,31);(1,3)(2,5)(4,8)(6,7)(17,20,24,22)(18,25,23,30)(19,28,26,29)(21,32,27,31);(1,2)(3,6)(4,7)(5,8)(17,21,24,27)(18,26,23,19)(20,31,22,32)(25,28,30,29);(1,2)(3,5)(4,7)(6,8)(9,11,13,12)(10,15,14,16)(17,22,24,20)(18,25,23,30)(19,29,26,28)(21,32,27,31);(1,3)(2,6)(4,8)(5,7)(9,12,13,11)(10,16,14,15)(17,23,24,18)(19,27,26,21)(20,30,22,25)(28,31,29,32)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=g,f^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, 7567827);