#7641971

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4774 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7549167);