#7672291

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	#39471	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=gh,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=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, 7579487);