#8475823

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	#53447	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(20,28)(24,29)(27,32)(30,31)(34,40)(36,44)(37,45)(41,48);(2,3)(7,9)(8,10)(14,15)(17,18,22,26)(19,21,25,23)(20,29,28,24)(27,31,32,30)(33,34)(35,41)(36,42)(37,46)(38,44)(39,40)(43,45)(47,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,35,39,47)(34,41,40,48)(36,45,44,37)(38,46,42,43);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,21)(20,27)(22,25)(23,26)(24,30)(28,32)(29,31)(33,36)(34,42)(35,45)(37,47)(38,40)(39,44)(41,43)(46,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,37,39,45)(34,43,40,46)(35,36,47,44)(38,48,42,41);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,25)(19,26)(20,30)(22,23)(24,32)(27,29)(28,31)(33,38,39,42)(34,44,40,36)(35,46,47,43)(37,48,45,41)

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=h,e^2=g,f^2=g,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, 8383019);