#7664984

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	#38426	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4852 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gi,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, 7572180);