#7650179

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4446 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7557375);