#10424685

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	#49754	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4175 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:

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

Polycyclic presentation:

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

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, 10331881);