#7653763

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4715 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:

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

Polycyclic presentation:

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

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, 7560959);