#8097109

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 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)(3,9)(4,8)(5,13)(6,10)(7,15)(11,14)(12,16)(17,18,23,28)(19,22,27,24)(20,25,30,32)(21,31,29,26)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,3)(2,5)(4,12)(6,13)(7,16)(8,14)(9,10)(11,15)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,31)(26,32)(35,41)(39,42)(43,47)(46,48);(1,4)(2,7)(3,11)(5,14)(6,15)(8,10)(9,16)(12,13)(17,20,23,30)(18,25,28,32)(19,29,27,21)(22,26,24,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,5,6,9)(2,3,10,13)(4,14,15,16)(7,11,8,12)(17,21,23,29)(18,26,28,31)(19,30,27,20)(22,25,24,32);(1,4)(2,8)(3,12)(5,14)(6,15)(7,10)(9,16)(11,13)(17,19)(18,22)(20,29)(21,30)(23,27)(24,28)(25,26)(31,32)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,3)(2,9)(4,11)(5,10)(6,13)(7,16)(8,14)(12,15)(17,22)(18,27)(19,28)(20,26)(21,25)(23,24)(29,32)(30,31)(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,d^2=g,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, 8004305);