#7631133

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	#36504	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5062 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:

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

Polycyclic presentation:

2^9|b^2=gi,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, 7538329);