#8095228

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4117 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(18,19)(21,29)(23,25)(28,32)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,47)(46,48);(3,9)(7,13)(10,15)(14,16)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(34,35)(39,41)(40,42)(46,47);(2,6)(7,13)(8,11)(14,16)(20,27)(23,25)(28,32)(30,31)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,40)(35,42)(36,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=g,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002424);