#8202257

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	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

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

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, 8109453);