#10217012

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3778 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)(10,16)(14,15)(17,18,22,23)(19,20,24,28)(21,29,31,25)(26,30,32,27)(34,39)(35,42)(41,48)(43,44);(2,7)(4,15)(8,11)(10,13)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,30)(31,32)(33,34,38,39)(35,40,42,37)(36,41,46,48)(43,45,44,47);(2,7)(3,12)(9,14)(10,13)(18,23)(19,24)(25,29)(26,32);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,30)(31,32)(33,35)(34,40)(36,44)(37,39)(38,42)(41,47)(43,46)(45,48);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,20)(18,24)(19,23)(21,30)(22,28)(25,26)(27,31)(29,32)(33,36,38,46)(34,41,39,48)(35,43,42,44)(37,47,40,45);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21,22,31)(18,25,23,29)(19,26,24,32)(20,27,28,30)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,44,48,43)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=hi,e^2=ghi,f^2=gi,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, 10124208);