#7658871

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	#41982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4672 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566067);