#8617765

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	#39552	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8524961);