#10481068

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	#52365	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3844 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=gi,e^2=g,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, 10388264);