#8751694

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4389 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gh,d^2=gi,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658890);