#7630594

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	#35212	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4830 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)(10,13)(14,16)(17,18)(19,20)(21,25)(22,30)(23,24)(26,31)(27,32)(28,29);(1,2)(3,7)(4,8)(5,6)(9,10,12,13)(11,14,15,16)(18,24)(19,21)(26,30)(27,29);(1,2)(3,7)(4,8)(5,6)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,32,30,28);(17,20,23,25)(18,19,24,21)(22,28,31,32)(26,27,30,29);(1,3,5,4)(2,7,6,8)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,32,30,28);(1,4,5,3)(2,8,6,7)(9,11,12,15)(10,14,13,16)(17,22,23,31)(18,26,24,30)(19,27,21,29)(20,28,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537790);