#8762003

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	#47996	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8669199);