#7864666

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	#35353	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4879 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=gh,e^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, 7771862);