#8928019

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3851 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,34,39,40)(35,43,44,41)(36,47,48,37)(38,45,46,42);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,18,22,20)(19,23,27,25)(21,28,29,24)(26,31,32,30)(34,40)(35,44)(37,47)(42,45);(3,11)(4,12)(6,16)(8,13)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35,39,44)(34,41,40,43)(36,45,48,42)(37,46,47,38);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,36)(34,37)(35,45)(38,43)(39,48)(40,47)(41,46)(42,44);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,37)(34,36)(35,46)(38,44)(39,47)(40,48)(41,45)(42,43);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,38,39,46)(34,42,40,45)(35,37,44,47)(36,43,48,41)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8835215);