#8285441

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

3940 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)(4,13)(5,10)(6,7)(8,12)(9,15)(14,16)(17,18)(19,24)(20,22)(21,25)(23,27)(26,31)(28,29)(30,32)(33,34,39,42)(35,40,43,38)(36,37,41,47)(44,45,48,46);(2,7)(3,12)(9,13)(10,14)(17,19)(18,23)(20,24)(21,26)(22,27)(25,30)(28,31)(29,32)(33,35,39,43)(34,40,42,38)(36,44,41,48)(37,45,47,46);(2,7)(5,16)(8,11)(9,13)(21,29)(25,28)(26,32)(30,31)(35,43)(38,40)(44,48)(45,46);(1,3)(2,8)(4,14)(5,13)(6,12)(7,11)(9,16)(10,15)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,36)(34,37)(35,44)(38,46)(39,41)(40,45)(42,47)(43,48);(1,4,6,15)(2,9,7,13)(3,10,12,14)(5,11,16,8)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,37)(34,41)(35,45)(36,42)(38,44)(39,47)(40,48)(43,46);(1,5)(2,10)(3,13)(4,11)(6,16)(7,14)(8,15)(9,12)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38)(34,35)(36,46)(37,44)(39,40)(41,45)(42,43)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gh,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

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, 8192637);