#10040781

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3733 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9947977);