#8755105

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	#46868	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,8)(4,11)(5,9)(6,7)(10,15)(12,16)(13,14)(17,18)(19,24)(20,29)(21,22)(23,26)(25,32)(27,31)(28,30)(34,37)(35,41)(40,44)(43,47);(2,7)(4,8)(5,13)(10,16)(17,19,23,20)(18,24,26,29)(21,30,25,27)(22,28,32,31)(33,34,38,37)(35,42,41,39)(36,44,45,40)(43,46,47,48);(1,3,6,11)(2,8,7,4)(5,10,13,16)(9,15,14,12)(17,20,23,19)(18,24,26,29)(21,30,25,27)(22,31,32,28)(33,35,38,41)(34,39,37,42)(36,43,45,47)(40,46,44,48);(1,3)(2,4)(5,10)(6,11)(7,8)(9,12)(13,16)(14,15)(33,34,38,37)(35,39,41,42)(36,40,45,44)(43,46,47,48);(1,4)(2,3)(5,12)(6,8)(7,11)(9,10)(13,15)(14,16)(17,21,23,25)(18,22,26,32)(19,27,20,30)(24,31,29,28)(33,36,38,45)(34,40,37,44)(35,43,41,47)(39,46,42,48);(1,5,6,13)(2,9,7,14)(3,10,11,16)(4,12,8,15)(17,22)(18,25)(19,28)(20,31)(21,26)(23,32)(24,27)(29,30)(33,37,38,34)(35,42,41,39)(36,44,45,40)(43,48,47,46)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662301);