#7631170

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	#36430	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5192 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)(3,8)(10,14)(15,16)(17,18)(19,24)(20,22)(21,26)(23,25)(27,31)(28,29)(30,32);(1,2)(3,7)(4,8)(5,6)(9,10,13,14)(11,15,12,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31);(1,3,5,8)(2,7,6,4)(9,11,13,12)(10,15,14,16)(18,25)(19,20)(26,32)(27,28);(17,20,23,19)(18,22,25,24)(21,27,30,28)(26,31,32,29);(17,21,23,30)(18,26,25,32)(19,27,20,28)(22,29,24,31);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,16,14,15)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=i,e^2=i,f^2=g,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, 7538366);