#10481008

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	#51758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10388204);