#7674477

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4675 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581673);