#8893608

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	#47170	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800804);