#9778472

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9685668);