#8446007

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	#54823	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=gh,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, 8353203);