#8214540

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	#36318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4780 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8121736);