#8994328

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	#44259	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=gi,e^2=h,f^2=g,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, 8901524);