#8505959

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	#39144	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)(4,5)(6,7)(8,13)(9,14)(11,12)(15,16)(17,18,22,26)(19,23,29,24)(20,25,31,21)(27,28,30,32)(36,44)(40,45)(43,48)(46,47);(2,7)(4,14)(8,10)(11,16)(17,19,22,29)(18,23,26,24)(20,30,31,27)(21,28,25,32)(33,34)(35,39)(36,45)(37,41)(38,42)(40,44)(43,46)(47,48);(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,32)(30,31)(33,35)(34,37)(36,43)(38,41)(39,42)(40,46)(44,48)(45,47);(1,3,6,13)(2,8,7,10)(4,11,14,16)(5,15,9,12)(17,19)(18,23)(20,27)(21,32)(22,29)(24,26)(25,28)(30,31)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,48)(45,46);(1,4)(2,9)(3,11)(5,7)(6,14)(8,12)(10,15)(13,16)(17,20)(18,25)(19,27)(21,26)(22,31)(23,28)(24,32)(29,30)(33,36,38,44)(34,40,42,45)(35,43,41,48)(37,46,39,47);(1,5,6,9)(2,4,7,14)(3,12,13,15)(8,16,10,11)(17,21,22,25)(18,20,26,31)(19,28,29,32)(23,30,24,27)(33,37,38,39)(34,41,42,35)(36,46,44,47)(40,48,45,43)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413155);