#10324597

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	#48130	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=g,f^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, 10231793);