#404782

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=g,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311978);