#10247642

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

3771 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)(3,12)(4,13)(5,8)(6,10)(7,11)(9,15)(14,16)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(1,3,7,5)(2,8,11,12)(4,14,15,10)(6,9,16,13)(17,18,22,25)(19,20,27,23)(21,24,30,32)(26,28,29,31)(33,34)(35,39)(36,44)(37,42)(38,41)(40,46)(43,48)(45,47);(17,19)(18,23)(20,25)(21,26)(22,27)(24,31)(28,32)(29,30)(33,35,38,37)(34,39,41,42)(36,45,46,43)(40,48,44,47);(1,4,7,15)(2,9,11,13)(3,10,5,14)(6,12,16,8)(17,20,22,23)(18,19,25,27)(21,28,30,31)(24,26,32,29)(33,36,38,46)(34,40,41,44)(35,43,37,45)(39,47,42,48);(1,5,7,3)(2,8,11,12)(4,10,15,14)(6,9,16,13)(17,19)(18,20)(21,29)(22,27)(23,25)(24,31)(26,30)(28,32)(33,37,38,35)(34,39,41,42)(36,43,46,45)(40,48,44,47);(1,6,7,16)(2,10,11,14)(3,9,5,13)(4,12,15,8)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154838);