#8908352

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	#53509	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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)(18,23)(19,26)(25,31)(27,28)(33,34)(35,41)(36,46)(37,43)(38,42)(39,40)(44,48)(45,47);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,32)(28,29)(30,31)(33,35,40,37)(34,41,39,43)(36,45,38,44)(42,48,46,47);(3,11)(4,13)(7,14)(8,16)(19,26)(21,24)(27,28)(29,32)(33,36,40,38)(34,42,39,46)(35,44,37,45)(41,47,43,48);(1,4)(2,8)(3,12)(5,7)(6,13)(9,11)(10,16)(14,15)(17,19,22,26)(18,24,23,21)(20,28,30,27)(25,29,31,32)(33,37,40,35)(34,43,39,41)(36,44,38,45)(42,47,46,48);(1,5)(2,9)(3,8)(4,14)(6,15)(7,13)(10,12)(11,16)(17,20,22,30)(18,25,23,31)(19,27,26,28)(21,29,24,32)(33,38,40,36)(34,42,39,46)(35,45,37,44)(41,47,43,48);(1,3)(2,7)(4,12)(5,8)(6,11)(9,13)(10,14)(15,16)(17,21,22,24)(18,26,23,19)(20,29,30,32)(25,28,31,27)(33,39,40,34)(35,43,37,41)(36,42,38,46)(44,47,45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=gh,f^2=gh,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, 8815548);