#10504575

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

4034 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=g,e^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10411771);