#7734384

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	#43904	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4935 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)(9,10,13,12)(11,15,16,14)(17,18)(19,28)(20,25)(21,24)(22,26)(23,27)(29,32)(30,31);(2,7)(3,5)(9,11,13,16)(10,14,12,15)(17,19,23,21)(18,24,27,28)(20,26,31,29)(22,25,32,30);(1,3,6,5)(2,8,7,4)(9,10,13,12)(11,14,16,15)(17,20,23,31)(18,25,27,30)(19,29,21,26)(22,28,32,24);(17,20)(18,25)(19,26)(21,29)(22,28)(23,31)(24,32)(27,30);(1,4,6,8)(2,3,7,5)(9,10,13,12)(11,14,16,15)(17,21,23,19)(18,24,27,28)(20,29,31,26)(22,25,32,30);(1,5,6,3)(2,8,7,4)(9,12,13,10)(11,15,16,14)(17,22,23,32)(18,26,27,29)(19,30,21,25)(20,28,31,24)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7641580);