#8421802

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

6248 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8328998);