#10457927

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

4001 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18,23,28)(19,24,30,32)(20,21,26,25)(22,31,29,27)(33,34,38,40)(35,39,44,48)(36,37,45,47)(41,42,43,46);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(17,19,23,30)(18,24,28,32)(20,29,26,22)(21,27,25,31)(33,35)(34,39)(36,43)(37,46)(38,44)(40,48)(41,45)(42,47);(17,20)(18,25)(19,29)(21,28)(22,30)(23,26)(24,31)(27,32)(33,36)(34,37)(35,41)(38,45)(39,42)(40,47)(43,44)(46,48);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,20)(18,21)(19,29)(22,30)(23,26)(24,27)(25,28)(31,32)(33,37)(34,36)(35,42)(38,47)(39,41)(40,45)(43,48)(44,46);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,21)(18,26)(19,27)(20,28)(22,24)(23,25)(29,32)(30,31)(33,36)(34,37)(35,43)(38,45)(39,46)(40,47)(41,44)(42,48);(1,6,7,14)(2,9,10,16)(3,11,5,13)(4,12,8,15)(17,22,23,29)(18,27,28,31)(19,20,30,26)(21,24,25,32)(33,35)(34,39)(36,41)(37,42)(38,44)(40,48)(43,45)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,e^2=g,f^2=gh,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, 10365123);