#10328265

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	#54056	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34,40,44)(35,45,48,41)(36,37,38,42)(39,46,47,43);(2,3)(5,13)(7,9)(12,16)(19,27)(21,31)(23,32)(25,28)(33,35,40,48)(34,41,44,45)(36,43,38,46)(37,47,42,39);(4,11)(7,9)(12,16)(14,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36,40,38)(34,42,44,37)(35,46,48,43)(39,41,47,45);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,37)(34,36)(35,47)(38,44)(39,48)(40,42)(41,43)(45,46);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,38,40,36)(34,42,44,37)(35,46,48,43)(39,45,47,41);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,39,40,47)(34,43,44,46)(35,37,48,42)(36,41,38,45)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=h,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, 10235461);