#8458642

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	#39909	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4104 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)(19,28)(20,24)(21,25)(22,26)(23,27)(29,30)(31,32)(36,44)(40,47)(43,48)(45,46);(1,3)(2,8)(4,12)(5,13)(6,11)(7,14)(9,10)(15,16)(19,20)(24,28)(29,31)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(17,19,23,20)(18,24,27,28)(21,29,26,31)(22,30,25,32)(33,35)(34,37)(36,43)(38,41)(39,42)(40,45)(44,48)(46,47);(1,4,7,13)(2,9,6,15)(3,12,14,5)(8,10,11,16)(17,20,23,19)(18,24,27,28)(21,31,26,29)(22,30,25,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,48)(45,47);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,9,11,15)(17,21,23,26)(18,25,27,22)(19,29,20,31)(24,32,28,30)(33,36,38,44)(34,40,42,47)(35,43,41,48)(37,45,39,46);(1,6,7,2)(3,11,14,8)(4,9,13,15)(5,16,12,10)(17,22,23,25)(18,26,27,21)(19,30,20,32)(24,31,28,29)(33,37,38,39)(34,41,42,35)(36,45,44,46)(40,48,47,43)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=gh,e^2=gi,f^2=gi,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, 8365838);