#8912453

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	#37673	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4776 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:

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

Polycyclic presentation:

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

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, 8819649);