#8112549

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	#56402	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(21,29)(25,30)(26,31)(27,32)(33,34)(35,38)(36,41)(37,40)(39,44)(42,45)(43,47)(46,48);(2,6)(7,9)(8,12)(14,15)(17,18)(19,20)(21,25)(22,23)(24,28)(26,27)(29,30)(31,32)(34,35)(39,41)(40,42)(46,47);(3,10)(7,9)(11,16)(14,15)(18,23)(19,24)(25,30)(26,31)(33,34,38,35)(36,39,44,41)(37,40,45,42)(43,46,48,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19,22,24)(18,20,23,28)(21,26,29,31)(25,27,30,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,21)(18,25)(19,26)(20,27)(22,29)(23,30)(24,31)(28,32)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

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

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, 8019745);