#7660916

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	#39359	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4724 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,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, 7568112);