#8136896

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	#50979	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=g,f^2=h,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, 8044092);