#9815029

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	#2825	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3803 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=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, 9722225);