#8818256

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	#392	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=hi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8725452);