#7695809

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4245 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=hi,f^2=i,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, 7603005);