#7851445

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

5051 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:

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

Polycyclic presentation:

2^9|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, 7758641);