#8127467

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	#35071	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3996 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)(3,11)(4,5)(6,8)(7,12)(9,10)(13,16)(14,15)(17,18)(19,26)(20,23)(21,28)(22,25)(24,30)(27,29)(31,32)(33,34,39,44)(35,37,42,40)(36,45,48,41)(38,46,47,43);(1,2,7,12)(3,8,6,11)(4,5,10,9)(13,14,15,16)(18,20)(19,25)(24,28)(27,31)(33,35)(34,40)(36,43)(37,44)(38,41)(39,42)(45,47)(46,48);(1,3,7,6)(2,8,12,11)(4,15,10,13)(5,16,9,14)(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35)(34,37)(36,46)(38,41)(39,42)(40,44)(43,48)(45,47);(1,4,7,10)(2,9,12,5)(3,13,6,15)(8,16,11,14)(17,20,23,18)(19,26,25,22)(21,28,30,24)(27,32,31,29)(33,36,39,48)(34,41,44,45)(35,43,42,46)(37,47,40,38);(1,5,7,9)(2,10,12,4)(3,14,6,16)(8,15,11,13)(17,21,23,30)(18,24,20,28)(19,27,25,31)(22,29,26,32)(33,37)(34,42)(35,44)(36,47)(38,48)(39,40)(41,46)(43,45);(1,6,7,3)(2,11,12,8)(4,15,10,13)(5,16,9,14)(17,22,23,26)(18,25,20,19)(21,29,30,32)(24,31,28,27)(33,38,39,47)(34,43,44,46)(35,41,42,45)(36,37,48,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^c=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, 8034663);