#8158960

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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)(4,5,14,16)(11,15,13,12)(17,18)(19,25)(20,27)(21,29)(22,24)(23,30)(26,28)(31,32)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3,7,6)(2,8,10,9)(4,11,14,13)(5,12,16,15)(17,18,22,24)(19,25,27,20)(21,23,30,29)(26,31,32,28)(34,35)(39,41)(40,42)(46,47);(1,4)(2,5)(3,11)(6,13)(7,14)(8,12)(9,15)(10,16)(17,18,22,24)(19,25,27,20)(21,23,30,29)(26,31,32,28)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,5,7,16)(2,4,10,14)(3,12,6,15)(8,11,9,13)(17,19,22,27)(18,20,24,25)(21,26,30,32)(23,28,29,31)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,4)(2,5)(3,11)(6,13)(7,14)(8,12)(9,15)(10,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31);(1,6,7,3)(2,9,10,8)(4,13,14,11)(5,15,16,12)(17,21)(18,23)(19,26)(20,28)(22,30)(24,29)(25,31)(27,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,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,d^c=dgi,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, 8066156);