#10216675

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

3914 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:

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

Polycyclic presentation:

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

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, 10123871);