#8472408

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8379604);