#8137604

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	#37456	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4250 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^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, 8044800);