#8445847

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3992 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(17,18,23,27)(19,28,29,24)(20,25,30,32)(21,22,26,31)(33,34)(35,42)(36,46)(37,41)(38,39)(40,45)(43,44)(47,48);(1,3,7,14)(2,8,11,12)(4,13,5,6)(9,16,15,10)(19,29)(20,30)(24,28)(25,32)(34,39)(36,41)(40,42)(43,47);(1,4,7,5)(2,9,11,15)(3,13,14,6)(8,16,12,10)(17,19)(18,24)(20,26)(21,30)(22,25)(23,29)(27,28)(31,32)(33,35,38,45)(34,40,39,42)(36,43,41,47)(37,44,46,48);(1,5,7,4)(2,9,11,15)(3,6,14,13)(8,16,12,10)(17,20,23,30)(18,25,27,32)(19,26,29,21)(22,28,31,24)(33,36)(34,37)(35,43)(38,41)(39,46)(40,44)(42,48)(45,47);(1,6)(2,10)(3,4)(5,14)(7,13)(8,9)(11,16)(12,15)(17,21,23,26)(18,22,27,31)(19,30,29,20)(24,25,28,32);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,22)(18,26)(19,25)(20,24)(21,27)(23,31)(28,30)(29,32)(33,37)(34,41)(35,44)(36,39)(38,46)(40,47)(42,43)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=gh,e^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353043);