#8097049

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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)(4,9)(5,10)(6,7)(8,12)(13,15)(14,16)(17,18,22,26)(19,23,28,31)(20,24,27,30)(21,25,29,32)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(2,7)(3,12)(9,16)(10,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(35,41)(39,42)(43,47)(46,48);(1,3)(2,8)(4,13)(5,9)(6,12)(7,11)(10,14)(15,16)(17,19,22,28)(18,23,26,31)(20,21,27,29)(24,25,30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,3)(2,8)(4,13)(5,9)(6,12)(7,11)(10,14)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(1,4)(2,9)(3,13)(5,8)(6,14)(7,16)(10,12)(11,15)(17,20)(18,24)(19,21)(22,27)(23,25)(26,30)(28,29)(31,32)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,5)(2,10)(3,9)(4,11)(6,15)(7,13)(8,14)(12,16)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004245);