#8762102

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	#57197	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4118 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669298);