#445861

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

888 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)(5,6)(7,17,21,12)(8,22)(9,13)(10,16)(11,15)(14,25)(18,28,20,27)(19,33,35,36)(23,30)(24,31)(26,34,29,32);(5,7)(6,12)(8,19)(9,20)(10,18)(11,21)(13,28)(14,29)(15,17)(16,27)(22,36)(23,35)(24,34)(25,32)(26,31)(30,33);(6,13,15,16)(7,18,21,20)(8,23)(14,30,31,22)(19,34,35,32)(24,25)(26,29)(33,36);(2,4)(5,8,9,24,11,23,10,25)(6,14,16,30,15,31,13,22)(7,19,20,34,21,35,18,32)(12,26,27,36,17,29,28,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353057);