#427270

order	512
nilpotency class	3
isoclinism	isoclinic class of #417566
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1375 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)(5,6,11,16)(7,17,21,27)(8,13,24,31)(9,15,10,14)(12,19,28,20)(18,33,35,29)(22,36,34,26)(23,32,25,30);(5,7)(6,12)(8,22)(9,19)(10,20)(11,21)(13,29)(14,27)(15,17)(16,28)(18,23)(24,34)(25,35)(26,30)(31,33)(32,36);(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,30)(15,32)(16,31)(17,29)(19,34)(20,22)(21,35)(27,33)(28,36);(2,4)(7,19,21,20)(12,27,28,17)(18,34,35,22)(26,33,36,29)

Polycyclic presentation:

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

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, 334466);