#7779423

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

4791 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7686619);