#7646597

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	#42444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4719 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7553793);