#10504983

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	#56350	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10412179);