#8108044

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

3998 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=g,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,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8015240);