#8908325

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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)(3,9)(4,12)(5,11)(6,7)(8,13)(10,15)(14,16)(17,18,22,26)(19,23,27,30)(20,24,29,32)(21,25,31,28)(33,34)(35,41)(36,46)(37,43)(38,42)(39,40)(44,48)(45,47);(1,3,6,11)(2,5,7,9)(4,10,13,14)(8,15,12,16)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,35,40,37)(34,41,39,43)(36,45,38,44)(42,48,46,47);(2,7)(4,13)(5,9)(10,14)(19,27)(20,29)(23,30)(24,32)(33,36,40,38)(34,42,39,46)(35,44,37,45)(41,47,43,48);(1,4)(2,8)(3,10)(5,15)(6,13)(7,12)(9,16)(11,14)(17,20)(18,24)(19,28)(21,23)(22,29)(25,27)(26,32)(30,31)(33,37,40,35)(34,43,39,41)(36,44,38,45)(42,47,46,48);(1,3)(2,9)(4,14)(5,7)(6,11)(8,15)(10,13)(12,16)(17,21)(18,25)(19,24)(20,30)(22,31)(23,29)(26,28)(27,32)(33,38,40,36)(34,42,39,46)(35,45,37,44)(41,47,43,48);(1,5,6,9)(2,3,7,11)(4,15,13,16)(8,10,12,14)(17,19)(18,23)(20,28)(21,24)(22,27)(25,29)(26,30)(31,32)(33,39,40,34)(35,43,37,41)(36,42,38,46)(44,47,45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=gh,e^2=gh,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, 8815521);