#8107700

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

3975 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=hi,d^2=hi,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

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, 8014896);