#8463843

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	#54442	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(19,20)(23,24)(27,28)(30,31)(33,34,38,42)(35,43,44,39)(36,40,46,45)(37,41,47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(35,44)(36,46)(39,43)(40,45);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35,38,44)(34,39,42,43)(36,41,46,48)(37,45,47,40);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36)(34,40)(35,41)(37,39)(38,46)(42,45)(43,47)(44,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30);(1,2)(3,7)(4,8)(5,14)(6,9)(10,13)(11,15)(12,16)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,37)(34,41)(35,45)(36,43)(38,47)(39,46)(40,44)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=g,e^2=hi,f^2=hi,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, 8371039);