#7859310

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
prime radical	#59800	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4785 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,29)(28,32);(1,3,7,5)(2,4,8,6)(9,10)(11,13)(12,14)(15,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27);(1,4,7,6)(2,5,8,3)(17,18,22,26)(19,23,30,27)(20,24,29,31)(21,25,28,32);(1,3,7,5)(2,4,8,6)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31);(1,5,7,3)(2,4,8,6)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,32,27,25);(1,6,7,4)(2,5,8,3)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=gh,e^2=h,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, 7766506);