#8285461

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	#57468	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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)(3,11)(4,14)(5,8)(6,12)(7,9)(10,15)(13,16)(18,23)(19,24)(25,30)(26,31)(33,34,39,42)(35,40,43,38)(36,37,41,47)(44,45,48,46);(1,3,7,5)(2,8,9,11)(4,12,15,13)(6,14,16,10)(17,18,22,23)(19,20,24,28)(21,25,29,30)(26,27,31,32)(33,35,39,43)(34,40,42,38)(36,44,41,48)(37,45,47,46);(2,9)(6,16)(8,11)(10,14)(21,29)(25,30)(26,31)(27,32)(35,43)(38,40)(44,48)(45,46);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,31)(30,32)(33,36)(34,37)(35,44)(38,46)(39,41)(40,45)(42,47)(43,48);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,37)(34,41)(35,45)(36,42)(38,44)(39,47)(40,48)(43,46);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)(33,38)(34,35)(36,46)(37,44)(39,40)(41,45)(42,43)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8192657);