#9023609

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3911 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=g,e^2=g,f^2=gi,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, 8930805);