#444082

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

824 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)(6,12,16,15)(7,17,20,25)(14,21,30,29)(19,31,33,35);(5,6,10,12,11,16,9,15)(7,18,17,28,20,26,25,13)(8,21,23,30,24,29,22,14)(19,27,31,32,33,36,35,34);(5,7,11,20)(6,13)(8,19,24,33)(9,17,10,25)(12,26)(14,27)(15,18)(16,28)(21,34)(22,31,23,35)(29,32)(30,36);(2,4)(5,8,9,22,11,24,10,23)(6,14,12,21,16,30,15,29)(7,19,17,31,20,33,25,35)(13,27,26,34,28,36,18,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351278);