#8202427

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	#38208	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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)(4,12)(5,9)(6,7)(8,13)(11,14)(15,16)(17,18,23,26)(19,24,30,25)(20,31,21,22)(27,29,28,32)(33,34)(35,38)(36,43)(37,40)(39,44)(41,47)(42,46)(45,48);(2,7)(3,9)(4,13)(14,16)(17,19,23,30)(18,24,26,25)(20,28,21,27)(22,29,31,32)(33,34,37,40)(35,38,42,46)(36,39,44,43)(41,45,48,47);(2,7)(5,10)(8,12)(14,16)(17,20,23,21)(18,22,26,31)(19,27,30,28)(24,32,25,29)(33,35,37,42)(34,38,40,46)(36,41,44,48)(39,45,43,47);(1,3,6,9)(2,5,7,10)(4,11,13,15)(8,14,12,16)(17,21,23,20)(18,22,26,31)(19,28,30,27)(24,32,25,29)(33,36)(34,39)(35,41)(37,44)(38,45)(40,43)(42,48)(46,47);(1,4)(2,8)(3,11)(5,14)(6,13)(7,12)(9,15)(10,16)(17,19)(18,25)(20,27)(21,28)(22,29)(23,30)(24,26)(31,32)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,44)(47,48);(1,5)(2,9)(3,7)(4,14)(6,10)(8,15)(11,12)(13,16)(17,22)(18,20)(19,29)(21,26)(23,31)(24,28)(25,27)(30,32)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8109623);