#7650215

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

4514 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=h,e^2=ghi,f^2=g,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, 7557411);