#10324627

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 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)(3,4)(5,13)(6,7)(8,15)(9,12)(10,16)(11,14)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34,40,44)(35,45,47,41)(36,39,37,42)(38,43,46,48);(1,3,6,9)(2,4,7,12)(5,14,15,10)(8,16,13,11)(19,27)(21,31)(23,32)(25,28)(33,35)(34,41)(36,43)(37,48)(38,39)(40,47)(42,46)(44,45);(2,7)(3,9)(8,13)(10,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36,40,37)(34,42,44,39)(35,43,47,48)(38,41,46,45);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,14)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,37,40,36)(34,39,44,42)(35,43,47,48)(38,41,46,45);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,38,40,46)(34,43,44,48)(35,39,47,42)(36,45,37,41);(1,5)(2,8)(3,10)(4,11)(6,15)(7,13)(9,14)(12,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,39)(34,36)(35,46)(37,44)(38,47)(40,42)(41,48)(43,45)

Polycyclic presentation:

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

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, 10231823);