#7664983

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	#38042	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4705 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7572179);