#7652792

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	#42975	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4700 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=gi,e^2=hi,f^2=gh,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, 7559988);