#7875543

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

4746 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7782739);