#7656845

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	#35821	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4945 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^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, 7564041);