#8475117

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

4129 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 8382313);