#8193837

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=ghi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8101033);