#10366158

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

3742 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=hi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10273354);