#7652820

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	#35064	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4952 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:

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

Polycyclic presentation:

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

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, 7560016);