#7697296

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

5000 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)(10,14)(15,16)(17,18,23,22)(19,20,27,24)(21,25,29,30)(26,28,32,31);(9,10,13,14)(11,15,12,16)(17,19,23,27)(18,24,22,20)(21,26,29,32)(25,31,30,28);(2,3)(7,8)(17,20,23,24)(18,19,22,27)(21,28,29,31)(25,26,30,32);(1,3,6,2)(4,8,5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,19)(21,28)(22,27)(23,24)(25,26)(29,31)(30,32);(1,4,6,5)(2,7,3,8)(9,12,13,11)(10,16,14,15)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32);(1,5,6,4)(2,8,3,7)(17,22,23,18)(19,20,27,24)(21,25,29,30)(26,31,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=gh,e^2=gh,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, 7604492);