#10440018

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10347214);