#10457908

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	#55362	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3867 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10365104);