#8436569

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	#40486	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4924 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=gh,e^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, 8343765);