#8445717

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	#43649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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)(17,18,23,27)(19,28,29,24)(20,25,31,30)(21,22,26,32)(34,39)(35,41)(36,44)(43,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(19,29)(20,31)(24,28)(25,30)(33,34,38,39)(35,40,41,46)(36,45,44,37)(42,48,47,43);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,19,23,29)(18,24,27,28)(20,26,31,21)(22,30,32,25)(33,35)(34,40)(36,42)(37,43)(38,41)(39,46)(44,47)(45,48);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,20)(18,25)(19,26)(21,29)(22,24)(23,31)(27,30)(28,32)(33,36,38,44)(34,37,39,45)(35,42,41,47)(40,43,46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,21,23,26)(18,22,27,32)(19,20,29,31)(24,30,28,25);(1,2)(3,7)(4,8)(5,14)(6,9)(10,13)(11,15)(12,16)(17,22)(18,26)(19,30)(20,28)(21,27)(23,32)(24,31)(25,29)(33,37)(34,36)(35,43)(38,45)(39,44)(40,42)(41,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=g,e^2=i,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, 8352913);