#8285159

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	#39252	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4219 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)(3,11,12,8)(4,14,15,9)(5,10,16,13)(18,19)(23,25)(24,26)(30,31)(33,34)(35,39)(36,44)(37,38)(40,46)(41,43)(42,48)(45,47);(2,7)(3,12)(9,14)(10,13)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,34)(35,41)(36,40)(37,38)(39,43)(42,48)(44,46)(45,47);(2,7)(5,16)(8,11)(9,14)(21,29)(24,26)(27,32)(30,31)(34,38)(39,41)(40,44)(47,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,9,16,14)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,11,16,8)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36,37,46)(34,40,38,44)(35,42,43,45)(39,47,41,48);(1,5)(2,10)(3,9)(4,11)(6,16)(7,13)(8,15)(12,14)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,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, 8192355);