#10521343

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

4337 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)(3,10)(4,7)(5,8)(6,9)(11,15)(12,13)(14,16)(17,18,24,27)(19,28,22,25)(20,23,21,26)(29,30,31,32)(33,34,40,43)(35,41,45,47)(36,46,37,42)(38,39,48,44);(1,3,6,8)(2,5,9,10)(4,12,14,11)(7,15,16,13)(17,19,24,22)(18,25,27,28)(20,31,21,29)(23,30,26,32)(33,35,40,45)(34,41,43,47)(36,44,37,39)(38,42,48,46);(17,20,24,21)(18,26,27,23)(19,29,22,31)(25,30,28,32)(33,36,40,37)(34,42,43,46)(35,44,45,39)(38,41,48,47);(1,4,6,14)(2,7,9,16)(3,11,8,12)(5,13,10,15)(17,21,24,20)(18,23,27,26)(19,29,22,31)(25,30,28,32)(33,37,40,36)(34,42,43,46)(35,39,45,44)(38,41,48,47);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,15)(14,16)(17,22,24,19)(18,25,27,28)(20,31,21,29)(23,32,26,30)(33,38)(34,39)(35,42)(36,41)(37,47)(40,48)(43,44)(45,46);(1,5)(2,8)(3,9)(4,13)(6,10)(7,12)(11,16)(14,15)(17,23)(18,20)(19,30)(21,27)(22,32)(24,26)(25,31)(28,29)(33,39)(34,38)(35,36)(37,45)(40,44)(41,42)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=ghi,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, 10428539);