#8908110

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	#35420	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8815306);