#10328326

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3890 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=h,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, 10235522);