#7699106

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

4807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606302);