#8119625

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	#52451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4006 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)(3,10)(4,8)(5,7)(6,13)(9,14)(11,15)(12,16)(20,28)(24,29)(26,32)(30,31)(33,34)(35,43)(36,40)(37,38)(39,42)(41,48)(44,47)(45,46);(3,4)(8,10)(11,12)(15,16)(18,22)(23,25)(24,29)(30,31)(33,35,39,36)(34,40,42,43)(37,44,48,46)(38,45,41,47);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,36,39,35)(34,40,42,43)(37,46,48,44)(38,45,41,47);(1,4,7,3)(2,8,5,10)(6,12,14,11)(9,15,13,16)(33,37)(34,41)(35,44)(36,46)(38,42)(39,48)(40,47)(43,45);(1,5,7,2)(3,8,4,10)(6,13,14,9)(11,15,12,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,38,39,41)(34,37,42,48)(35,45,36,47)(40,44,43,46);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,41)(35,44)(36,46)(38,42)(39,48)(40,47)(43,45)

Polycyclic presentation:

2^9|b^2=g,c^2=gh,d^2=h,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026821);