#8933201

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	#58991	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

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, 8840397);