#7648734

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	#38077	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4807 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

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, 7555930);