#8445319

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	#39218	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,15)(14,16)(20,28)(24,31)(27,32)(29,30)(33,34,40,37)(35,41,38,47)(36,45,44,42)(39,43,46,48);(1,3,8,6)(2,9,12,11)(4,13,5,14)(7,15,10,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,35,40,38)(34,41,37,47)(36,39,44,46)(42,48,45,43);(1,4,8,5)(2,10,12,7)(3,13,6,14)(9,16,11,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,42)(35,39)(37,45)(38,46)(40,44)(41,48)(43,47);(1,5,8,4)(2,10,12,7)(3,14,6,13)(9,16,11,15)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,37,40,34)(35,41,38,47)(36,45,44,42)(39,48,46,43);(1,6,8,3)(2,11,12,9)(4,14,5,13)(7,16,10,15)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,38,40,35)(34,41,37,47)(36,46,44,39)(42,48,45,43);(1,7,8,10)(2,5,12,4)(3,15,6,16)(9,14,11,13)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,39,40,46)(34,43,37,48)(35,44,38,36)(41,42,47,45)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=gh,e^2=ghi,f^2=gh,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, 8352515);