#8285519

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	#55340	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4043 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,d^2=h,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, 8192715);