#10158432

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	#55417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4124 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,d^2=g,e^2=i,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, 10065628);