#7631148

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5164 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)(10,15)(11,16)(17,18)(19,25)(20,26)(21,22)(23,24)(27,32)(28,29)(30,31);(9,10,14,15)(11,12,16,13)(18,24)(19,21)(26,31)(27,28);(1,2,4,5)(3,6,7,8)(9,11)(10,12)(13,15)(14,16)(17,19,23,21)(18,25,24,22)(20,28,30,27)(26,29,31,32);(9,12,14,13)(10,16,15,11)(17,20,23,30)(18,26,24,31)(19,27,21,28)(22,29,25,32);(9,12,14,13)(10,16,15,11)(17,21,23,19)(18,22,24,25)(20,27,30,28)(26,32,31,29);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,11,15,16)(17,22)(18,19)(20,29)(21,24)(23,25)(26,27)(28,31)(30,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538344);