#8994372

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=i,e^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, 8901568);