#8422189

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	#36263	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4706 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,e^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, 8329385);