#444619

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1103 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)(5,6,9,16,11,15,10,14)(7,17,20,28,21,12,19,29)(8,22,25,13,24,31,26,32)(18,34,23,27,35,30,36,33);(5,7)(6,12)(8,23)(9,19)(10,20)(11,21)(13,30)(14,29)(15,17)(16,28)(18,25)(22,33)(24,36)(26,35)(27,31)(32,34);(5,8,9,25,11,24,10,26)(6,13,14,22,15,32,16,31)(7,18,20,23,21,35,19,36)(12,27,28,34,17,33,29,30);(2,4)(6,14,15,16)(8,24)(12,28,17,29)(13,31,32,22)(18,35)(23,36)(25,26)(27,30,33,34)

Polycyclic presentation:

2^9|a^2=g,c^2=eg,d^2=e,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, 351815);