#10196122

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	#43265	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4093 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)(17,18,23,25)(19,24,28,32)(20,21,26,31)(22,30,29,27)(33,34,38,41)(35,37,43,39)(36,40,44,47)(42,45,48,46);(2,7)(4,8)(9,13)(12,14)(17,19,23,28)(18,24,25,32)(20,29,26,22)(21,27,31,30)(36,44)(40,47)(42,48)(45,46);(2,7)(3,10)(9,13)(11,16)(18,25)(19,28)(20,26)(27,30)(33,35,38,43)(34,39,41,37)(36,42,44,48)(40,46,47,45);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,20)(18,21)(19,29)(22,28)(23,26)(24,27)(25,31)(30,32)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,48)(46,47);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,21)(18,26)(19,30)(20,25)(22,32)(23,31)(24,29)(27,28)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,22,23,29)(18,27,25,30)(19,20,28,26)(21,24,31,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,d^2=g,e^2=g,f^2=gi,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, 10103318);