#478601

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,11,16,10,14,27,15)(7,17,20,28,19,9,13,26)(8,21,24,12,25,31,35,32)(18,34,22,23,36,29,30,33);(5,7,10,19)(6,9,14,17)(8,22)(11,20,27,13)(12,29)(15,28,16,26)(18,35)(21,23)(24,36)(25,30)(31,33)(32,34);(5,8,11,24,10,25,27,35)(6,12,15,21,14,32,16,31)(7,18,13,30,19,36,20,22)(9,23,26,29,17,33,28,34);(2,4)(5,9,10,17)(6,13,14,20)(7,16,19,15)(8,23,25,33)(11,28,27,26)(12,30,32,22)(18,31,36,21)(24,34,35,29)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385797);