#7642635

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	#35001	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5334 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)(3,8)(4,7)(5,6)(9,10,14,12)(11,16,13,15)(17,18,22,23)(19,21,25,30)(20,24,29,31)(26,28,27,32);(9,11,14,13)(10,15,12,16)(18,23)(19,25)(24,31)(26,27);(2,6)(7,8)(10,12)(15,16)(17,19)(18,21)(20,27)(22,25)(23,30)(24,32)(26,29)(28,31);(1,3,5,4)(2,7,6,8)(9,11,14,13)(10,15,12,16);(1,2)(3,7)(4,8)(5,6)(9,12,14,10)(11,16,13,15)(17,20,22,29)(18,24,23,31)(19,26,25,27)(21,28,30,32);(1,4,5,3)(2,8,6,7)(9,13,14,11)(10,16,12,15)(17,21)(18,19)(20,28)(22,30)(23,25)(24,26)(27,31)(29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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^c=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, 7549831);