#8755142

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	#54873	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3827 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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 8662338);