#10121192

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	#57474	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4091 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=hi,e^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, 10028388);