#8436622

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	#52718	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4090 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8343818);