#8295561

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8202757);