#7656861

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	#37440	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5018 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,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, 7564057);