#8136899

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	#55566	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4113 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)(3,10)(4,13)(5,8)(6,14)(7,9)(11,12)(15,16)(20,28)(24,31)(26,32)(29,30)(33,34,38,39)(35,42,37,40)(36,44,46,41)(43,47,45,48);(1,3,7,5)(2,8,9,10)(4,11,14,15)(6,12,13,16)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(34,39)(40,42)(41,44)(47,48);(17,19)(18,22)(20,26)(21,27)(23,25)(24,29)(28,32)(30,31)(33,35,38,37)(34,40,39,42)(36,43,46,45)(41,47,44,48);(1,4)(2,6)(3,11)(5,15)(7,14)(8,12)(9,13)(10,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,34)(35,40)(36,44)(37,42)(38,39)(41,46)(43,48)(45,47);(1,5,7,3)(2,8,9,10)(4,11,14,15)(6,16,13,12)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,31)(30,32)(33,36,38,46)(34,41,39,44)(35,43,37,45)(40,47,42,48);(1,6)(2,4)(3,12)(5,16)(7,13)(8,11)(9,14)(10,15)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37,38,35)(34,42,39,40)(36,45,46,43)(41,48,44,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=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, 8044095);