#10202100

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=ghi,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, 10109296);