#8429650

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	#36152	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6168 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8336846);