#8196398

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	#38909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4746 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8103594);