#8741725

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	#59005	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4650 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=h,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, 8648921);