#8139494

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	#49852	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3969 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:

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

Polycyclic presentation:

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

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, 8046690);