#8505076

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=ghi,d^2=ghi,e^2=g,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, 8412272);