#7685103

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

4756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592299);