#8115501

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

3839 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,23,20)(19,24,26,22)(21,28,31,25)(27,30,29,32)(33,34)(35,40)(36,37)(38,48)(39,42)(41,46)(43,47)(44,45);(2,6)(7,9)(8,12)(14,15)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(3,10)(7,9)(11,16)(14,15)(18,20)(19,26)(25,28)(27,29)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20,23,18)(19,24,26,22)(21,28,31,25)(27,32,29,30)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,30,24,32)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,24)(18,26,20,19)(21,30,31,32)(25,29,28,27)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022697);