#7631247

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	C₂×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5132 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^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, 7538443);