#8475084

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

3899 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=gh,f^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, 8382280);