#8912294

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	#57288	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,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, 8819490);