#8934600

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	#59124	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5000 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)(9,10)(11,13)(12,16)(14,15)(17,18)(19,23)(20,22)(21,24)(25,27)(26,30)(28,29)(31,32);(1,3,5,8)(2,6,7,4)(10,15)(11,16)(18,20)(19,25)(24,28)(26,31);(1,3)(2,6)(4,7)(5,8)(9,11,14,16)(10,12,15,13)(19,25)(21,29)(23,27)(24,28);(2,7)(3,8)(9,12,14,13)(10,11,15,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32);(1,4)(2,3)(5,6)(7,8)(9,13,14,12)(10,16,15,11)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=h,e^2=gh,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, 8841796);