#8107699

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8014895);