#8445997

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=h,e^2=i,f^2=g,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, 8353193);