#8434811

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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:

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

Polycyclic presentation:

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

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, 8342007);