#7656987

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4648 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)(10,13)(11,15)(17,18,23,22)(19,28,30,24)(20,27,31,25)(21,26,29,32);(1,3,5,8)(2,6,4,7)(10,13)(14,16)(17,19,23,30)(18,24,22,28)(20,29,31,21)(25,32,27,26);(9,10,12,13)(11,14,15,16)(17,20)(18,25)(19,29)(21,30)(22,27)(23,31)(24,32)(26,28);(9,11,12,15)(10,14,13,16)(17,21,23,29)(18,26,22,32)(19,20,30,31)(24,25,28,27);(1,4,5,2)(3,7,8,6)(17,20)(18,27)(19,29)(21,30)(22,25)(23,31)(24,26)(28,32);(17,22,23,18)(19,28,30,24)(20,27,31,25)(21,32,29,26)

Polycyclic presentation:

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

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, 7564183);