#8436698

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	#53689	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=g,e^2=gh,f^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, 8343894);