#7649943

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

4842 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7557139);