#8762028

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	#44442	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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669224);