#7851443

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

4903 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)(19,27)(20,25)(21,28)(22,26)(23,24)(29,31)(30,32);(1,2)(3,6)(4,7)(5,8)(9,10,13,14)(11,15,12,16)(17,19,23,28)(18,21,24,27)(20,22,29,32)(25,30,31,26);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,15,14,16)(18,24)(19,28)(20,29)(26,30);(17,20,23,29)(18,25,24,31)(19,22,28,32)(21,30,27,26);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,21,23,27)(18,19,24,28)(20,30,29,26)(22,31,32,25);(1,2)(3,5)(4,7)(6,8)(9,12,13,11)(10,16,14,15)(17,22)(18,26)(19,29)(20,28)(21,25)(23,32)(24,30)(27,31)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,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, 7758639);