#8165152

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8072348);