#8374175

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

3848 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,e^2=h,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,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, 8281371);