#7685080

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

5432 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=i,e^2=ghi,f^2=i,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, 7592276);