#7917797

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

4955 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)(3,7)(4,6)(5,8)(10,12)(14,15)(17,18,22,25)(19,26,21,23)(20,28,30,24)(27,31,29,32);(1,3)(2,5)(4,8)(6,7)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,30)(29,31);(2,6)(5,7)(9,10,13,12)(11,15,16,14)(17,19,22,21)(18,23,25,26)(20,29,30,27)(24,32,28,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,12,15)(17,20,22,30)(18,24,25,28)(19,27,21,29)(23,31,26,32);(1,3)(2,5)(4,8)(6,7)(9,12,13,10)(11,15,16,14)(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,30)(29,32);(17,21,22,19)(18,23,25,26)(20,29,30,27)(24,31,28,32)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=ghi,e^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7824993);