#8479549

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386745);