#10444899

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

4039 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)(3,8,4,5)(6,9,14,16)(11,15,12,13)(17,18)(19,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32)(33,34,39,36)(35,40,37,42)(38,41,46,44)(43,47,45,48);(1,3,7,4)(2,5,10,8)(6,11,14,12)(9,13,16,15)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,48)(45,46);(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32)(33,34,39,36)(35,42,37,40)(38,41,46,44)(43,48,45,47);(1,4,7,3)(2,8,10,5)(6,11,14,12)(9,13,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32)(33,36,39,34)(35,42,37,40)(38,41,46,44)(43,47,45,48);(1,5,7,8)(2,4,10,3)(6,13,14,15)(9,12,16,11)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,37,39,35)(34,40,36,42)(38,45,46,43)(41,47,44,48);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,20,21,29)(18,24,25,28)(19,26,27,32)(22,30,23,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=gi,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, 10352095);