#8095127

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	#34737	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4830 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=h,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002323);