#8115579

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	#42977	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4216 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8022775);