#7709420

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	#36967	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4984 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)(3,5,4,7)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,27,28,24)(21,25,31,29)(22,30,32,26);(1,3,6,4)(2,7,8,5)(9,11,13,16)(10,14,12,15)(17,19)(18,24)(20,27)(21,22)(23,28)(25,26)(29,30)(31,32);(2,8)(3,4)(19,28)(21,31)(24,27)(25,29);(1,2,6,8)(3,7,4,5)(9,12,13,10)(11,15,16,14)(17,20,23,18)(19,27,28,24)(21,29,31,25)(22,30,32,26);(1,4,6,3)(2,5,8,7)(17,21)(18,25)(19,22)(20,29)(23,31)(24,26)(27,30)(28,32);(1,5)(2,3)(4,8)(6,7)(17,22)(18,26)(19,21)(20,30)(23,32)(24,25)(27,29)(28,31)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,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, 7616616);