#7631284

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	#35292	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5176 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=h,e^2=g,f^2=i,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, 7538480);