#436274

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 343470);