#7806809

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

4967 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)(9,10)(11,15)(12,14)(13,16)(17,18)(19,25)(20,26)(21,24)(22,23)(27,32)(28,29)(30,31);(1,2,5,6)(3,7,4,8)(17,19,22,26)(18,20,23,25)(21,29,30,27)(24,32,31,28);(1,3,5,4)(2,7,6,8)(9,10,12,14)(11,13,16,15)(18,23)(19,26)(21,30)(28,32);(9,10)(11,13)(12,14)(15,16)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,30)(31,32);(9,11)(10,13)(12,16)(14,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32);(1,4,5,3)(2,8,6,7)(17,21,22,30)(18,24,23,31)(19,27,26,29)(20,28,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7714005);