#8506042

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4009 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413238);