#10202072

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	#40139	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3996 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=h,d^2=hi,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, 10109268);