#9058512

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	#44153	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8965708);