#8110632

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	#60048	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4170 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=hi,e^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017828);