#7694254

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

4860 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7601450);