#7630856

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	#36029	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5188 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:

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

Polycyclic presentation:

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

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, 7538052);