#8617799

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

3869 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8524995);