#8174430

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	#59228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=ghi,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi

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, 8081626);