#7851446

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

5013 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7758642);