#9001093

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	#55097	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3914 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8908289);