#7696102

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

4694 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=g,e^2=ghi,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, 7603298);