#8898572

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	#47129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 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)(3,5)(4,11)(6,14)(7,9)(8,15)(10,12)(13,16)(17,18)(19,25)(20,26)(21,29)(22,23)(24,30)(27,28)(31,32)(33,34,39,43)(35,38,45,40)(36,46,44,41)(37,42,47,48);(1,3,7,11)(2,5,9,4)(6,10,15,16)(8,13,14,12)(18,23)(19,26)(24,29)(27,32)(33,35)(34,40)(36,37)(38,43)(39,45)(41,42)(44,47)(46,48);(1,4,7,5)(2,3,9,11)(6,13,15,12)(8,16,14,10)(17,19)(18,20)(21,27)(22,26)(23,25)(24,28)(29,31)(30,32)(33,36,39,44)(34,41,43,46)(35,37,45,47)(38,48,40,42);(1,5,7,4)(2,3,9,11)(6,13,15,12)(8,10,14,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,35)(34,40)(36,37)(38,43)(39,45)(41,42)(44,47)(46,48);(1,3)(2,5)(4,9)(6,10)(7,11)(8,13)(12,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,37)(34,42)(35,44)(36,45)(38,41)(39,47)(40,46)(43,48);(1,6)(2,8)(3,10)(4,12)(5,13)(7,15)(9,14)(11,16)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,38,39,40)(34,35,43,45)(36,42,44,48)(37,41,47,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=gh,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, 8805768);