#8110501

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	#46287	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4162 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=ghi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239255
maximal normal subgroup of	#72239255

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8017697);