#478476

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

829 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)(5,6,10,15)(7,16)(8,13,24,29)(9,12,23,14)(11,19)(17,27)(18,33)(20,26)(21,31)(22,28,35,30)(25,32)(34,36);(5,7,10,20)(6,11)(8,21,24,18)(9,17,23,19)(12,16)(13,25)(14,26)(15,27)(22,34,35,32)(28,31)(29,36)(30,33);(6,12)(7,17)(8,22)(9,23)(13,29)(14,15)(16,26)(18,21)(19,20)(24,35)(25,31)(33,36);(2,4)(5,8,23,35,10,24,9,22)(6,13,14,30,15,29,12,28)(7,18,19,34,20,21,17,32)(11,25,26,33,27,36,16,31)

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,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385672);