#8712865

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	#44001	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4855 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8620061);