#8898632

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

3855 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805828);