#8762025

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4288 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)(3,11)(4,9)(5,10)(6,7)(8,12)(13,14)(15,16)(17,18,22,23)(19,25,27,20)(21,29,30,24)(26,28,32,31)(33,34)(35,38)(36,44)(37,40)(39,45)(41,43)(42,47)(46,48);(2,7)(5,15)(8,11)(9,13)(18,23)(20,25)(21,30)(26,32)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(3,12)(4,14)(8,11)(9,13)(17,19,22,27)(18,20,23,25)(21,26,30,32)(24,28,29,31)(33,34,37,40)(35,41,43,38)(36,39,45,44)(42,47,48,46);(1,3)(2,8)(4,10)(5,13)(6,12)(7,11)(9,15)(14,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,31)(30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,13)(8,15)(12,16)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,32)(30,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,5,6,15)(2,10,7,16)(3,13,12,9)(4,11,14,8)(17,21,22,30)(18,24,23,29)(19,26,27,32)(20,28,25,31)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,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, 8669221);