#10171823

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gh,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10079019);