#10102217

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	#50822	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10009413);