#8109899

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4048 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)(3,4)(5,13)(6,7)(8,11)(9,14)(10,15)(12,16)(21,29)(25,30)(26,31)(28,32)(33,34,37,38)(35,41,43,39)(36,44,45,40)(42,46,48,47);(2,7)(4,8)(9,13)(12,15)(18,23)(20,27)(25,30)(28,32)(34,38)(36,45)(39,41)(42,48);(17,18,22,23)(19,20,24,27)(21,25,29,30)(26,28,31,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,13)(14,16)(17,19,22,24)(18,20,23,27)(21,26,29,31)(25,28,30,32)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47);(1,4,6,8)(2,3,7,11)(5,12,14,15)(9,10,13,16)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,30)(29,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,46)(44,45)(47,48);(1,5,6,14)(2,9,7,13)(3,10,11,16)(4,12,8,15)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017095);