#8928616

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

4076 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:

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

Polycyclic presentation:

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

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, 8835812);