#461127

order	512
nilpotency class	3
isoclinism	isoclinic class of #459444
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

897 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)(5,6,11,14,9,15,10,24)(7,16,20,28,19,12,17,26)(8,21,25,30,22,13,23,29)(18,27,32,36,34,31,33,35);(5,7,9,19)(6,12)(8,18,22,34)(10,17,11,20)(13,27)(14,26)(15,16)(21,31)(23,33,25,32)(24,28)(29,36)(30,35);(7,17)(10,11)(12,26)(14,24)(16,28)(18,33)(19,20)(23,25)(27,36)(29,30)(31,35)(32,34);(2,4)(5,8,9,22)(6,13,15,21)(7,18,19,34)(10,25,11,23)(12,27,16,31)(14,30,24,29)(17,32,20,33)(26,35,28,36)

Polycyclic presentation:

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

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, 368323);