#8118621

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	#54588	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3909 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8025817);