#7655582

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	#34575	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4870 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)(10,13)(11,15)(17,18)(19,26)(20,28)(21,23)(22,25)(24,30)(27,31)(29,32);(1,2,4,5)(3,6,7,8)(10,13)(14,16)(17,19,22,21)(18,23,25,26)(20,27,30,29)(24,31,28,32);(9,10)(11,14)(12,13)(15,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,30)(29,31);(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,28)(26,32);(1,3,4,7)(2,6,5,8)(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,30)(29,32);(17,21,22,19)(18,23,25,26)(20,29,30,27)(24,31,28,32)

Polycyclic presentation:

2^9|b^2=g,e^2=ghi,f^2=hi,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, 7562778);