#10195775

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	#41318	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18,22,25)(19,26,21,23)(20,24,29,32)(27,31,28,30)(33,34,39,43)(35,40,45,38)(36,41,44,48)(37,42,46,47);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(19,21)(23,26)(27,28)(30,31)(33,35)(34,40)(36,46)(37,44)(38,43)(39,45)(41,47)(42,48);(2,4)(3,10)(9,12)(11,16)(17,19,22,21)(18,23,25,26)(20,28,29,27)(24,31,32,30)(33,36,39,44)(34,41,43,48)(35,37,45,46)(38,47,40,42);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20,22,29)(18,24,25,32)(19,27,21,28)(23,30,26,31)(33,35)(34,40)(36,37)(38,43)(39,45)(41,42)(44,46)(47,48);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21,22,19)(18,23,25,26)(20,27,29,28)(24,31,32,30)(33,37)(34,42)(35,44)(36,45)(38,41)(39,46)(40,48)(43,47);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,18)(19,23)(20,24)(21,26)(22,25)(27,30)(28,31)(29,32)(33,38,39,40)(34,35,43,45)(36,47,44,42)(37,48,46,41)

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=h,e^2=gh,f^2=ghi,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, 10102971);