#473494

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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	#2782
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

673 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 380690);