#8127499

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3848 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(21,29)(23,30)(26,32)(28,31)(33,34,40,38)(35,42,39,37)(36,41,46,48)(43,44,45,47);(2,3)(7,9)(8,10)(14,15)(17,18,22,24)(19,25,27,20)(21,23,29,30)(26,31,32,28)(33,35,40,39)(34,37,38,42)(36,43,46,45)(41,47,48,44);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,22,27)(18,20,24,25)(21,26,29,32)(23,28,30,31)(33,36)(34,41)(35,43)(37,47)(38,48)(39,45)(40,46)(42,44);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,37)(34,39)(35,38)(36,44)(40,42)(41,43)(45,48)(46,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,31)(30,32)(33,38,40,34)(35,42,39,37)(36,41,46,48)(43,47,45,44);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,23)(19,26)(20,28)(22,29)(24,30)(25,31)(27,32)(33,39,40,35)(34,42,38,37)(36,45,46,43)(41,44,48,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=i,d^2=h,e^2=gh,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, 8034695);