#8139488

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	#55362	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^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, 8046684);