#7744926

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

4837 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:

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

Polycyclic presentation:

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

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, 7652122);