#8458582

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8365778);