#8750709

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657905);