#7806652

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

5688 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,e^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, 7713848);