#461274

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

848 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)(7,17,21,27)(8,14)(9,13)(10,15)(11,16)(12,20,25,18)(19,31,34,36)(22,28)(23,30)(24,29)(26,33,35,32);(5,7,11,21)(6,12)(8,19,24,34)(9,20,10,18)(13,17)(14,26)(15,27)(16,25)(22,33,23,32)(28,31)(29,35)(30,36);(6,13)(7,18)(8,22)(9,10)(12,25)(14,29)(15,16)(20,21)(23,24)(26,36)(31,35)(32,33);(2,4)(5,8,10,23,11,24,9,22)(6,14,15,30,16,29,13,28)(7,19,18,32,21,34,20,33)(12,26,27,36,25,35,17,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=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, 368470);